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%I M2948 #426 Apr 26 2024 02:44:50
%S 1,1,1,3,13,71,461,3447,29093,273343,2829325,31998903,392743957,
%T 5201061455,73943424413,1123596277863,18176728317413,311951144828863,
%U 5661698774848621,108355864447215063,2181096921557783605,46066653228356851631,1018705098450570562877
%N Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.
%C Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5].
%C Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3].
%C Related to number of subgroups of index n-1 in free group of rank 2 (i.e., maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.
%C Also the left border of triangle A144107, with row sums = n!. - _Gary W. Adamson_, Sep 11 2008
%C Hankel transform is A059332. Hankel transform of aerated sequence is A137704(n+1). - _Paul Barry_, Oct 07 2008
%C For every n, a(n+1) is also the moment of order n for the probability density function rho(x) = exp(x)/(Ei(1,-x)*(Ei(1,-x) + 2*I*Pi)) on the interval 0..infinity, with Ei the exponential-integral function. - _Groux Roland_, Jan 16 2009
%C Also (apparently), a(n+1) is the number of rooted hypermaps with n darts on a surface of any genus (see Walsh 2012). - _N. J. A. Sloane_, Aug 01 2012
%C Also recurrent sequence A233824 (for n > 0) in Panaitopol's formula for pi(x), the number of primes <= x. - _Jonathan Sondow_, Dec 19 2013
%C Also the number of mobiles (cyclic rooted trees) with an arrow from each internal vertex to a descendant of that vertex. - _Brad R. Jones_, Sep 12 2014
%C Up to sign, Möbius numbers of the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - _F. Chapoton_, Apr 29 2015
%C Also, a(n) is the number of distinct leaf matrices of complete non-ambiguous trees of size n. - _Daniel Chen_, Oct 23 2022
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 25, Example 20.
%D E. W. Bowen, Letter to N. J. A. Sloane, Aug 27 1976.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 84 (#25), 262 (#14) and 295 (#16).
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23, N_{n,2}.
%D I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in Handbook of Combinatorics, Vol. 2, edited by R. L. Graham et al., The MIT Press, Mass, 1995.
%D M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 22.
%D H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 1, Ex. 128; Vol. 2, 1999, see Problem 5.13(b).
%H Seiichi Manyama, <a href="/A003319/b003319.txt">Table of n, a(n) for n = 0..449</a> (first 102 terms from T. D. Noe)
%H Marcelo Aguiar and Frank Sottile, <a href="https://arxiv.org/abs/math/0203282">Structure of the Malvenuto-Reutenauer Hopf algebra of permutations</a>, arXiv:math/0203282 [math.CO], 2002-2005.
%H Marcelo Aguiar and Aaron Lauve, <a href="https://hal.inria.fr/hal-01229682">Convolution Powers of the Identity</a>, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, pp.1053-1064, 2013, DMTCS Proceedings. <hal-01229682>.
%H M. Aguiar and A. Lauve, <a href="https://web.archive.org/web/20210308062053/https://www.irif.fr/~chapuy//fpsac13websiteBackup/fpsac13/pdfAbstracts/dmAS0198.pdf">Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras</a>, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
%H M. Aguiar and A. Lauve, <a href="http://www.math.tamu.edu/~maguiar/adams.pdf">The characteristic polynomial of the Adams operators on graded connected Hopf algebras</a>, 2014.
%H Marcelo Aguiar and Swapneel Mahajan, <a href="http://mosaic.math.tamu.edu/~maguiar/hadamard.pdf">On the Hadamard product of Hopf monoids</a>, 2013.
%H Joerg Arndt, <a href="http://jjj.de/pub/arndt-rand-perm-thesis.pdf">Generating Random Permutations</a>, PhD thesis, Australian National University, Canberra, Australia, (2010).
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 281
%H Roland Bacher and Christophe Reutenauer, <a href="http://arxiv.org/abs/1511.00426">Number of right ideals and a q-analogue of indecomposable permutations</a>, arXiv preprint arXiv:1511.00426 [math.CO], 2015.
%H Paul Barry, <a href="https://arxiv.org/abs/1804.06801">A note on number triangles that are almost their own production matrix</a>, arXiv:1804.06801 [math.CO], 2018.
%H Julien Berestycki, Eric Brunet and Zhan Shi, <a href="http://arxiv.org/abs/1304.0246">How many evolutionary histories only increase fitness?</a>, arXiv preprint arXiv:1304.0246 [math.PR], 2013.
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1803.07727">A family of Bell transformations</a>, arXiv:1803.07727 [math.CO], 2018.
%H E. W. Bowen and N. J. A. Sloane, <a href="/A003319/a003319.pdf">Correspondence, 1976</a>
%H David Callan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Callan/callan91.html">Counting Stabilized-Interval-Free Permutations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
%H Peter Cameron's Blog, <a href="https://cameroncounts.wordpress.com/2011/04/09/the-symmetric-group-11/">The symmetric group, 11</a>, Posted 09/04/2011.
%H Mahir Bilen Can, Luke Nelson, and Kevin Treat, <a href="https://doi.org/10.54550/ECA2022V2S4PP7">A Catalanization Map on the Symmetric Group</a>, Enum. Comb. Appl. (2022) Vol. 2, No. 4, #S4PP7.
%H Daniel Chen and Sebastian Ohlig, <a href="https://arxiv.org/abs/2210.11117">Associated Permutations of Complete Non-Ambiguous Trees and Zubieta's Conjecture</a>, arXiv:2210.11117 [math.CO], 2022.
%H Fan Chung and R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194.
%H L. Comtet, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5619085m/f9.image">Sur les coefficients de l'inverse de la série formelle Sum n! t^n</a>, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
%H L. Comtet, <a href="/A003319/a003319_1.pdf">Series inversions</a>, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
%H Xinle Dai, Jordan Long, and Karen Yeats, <a href="https://arxiv.org/abs/2106.07494">Subdivergence-free gluings of trees</a>, arXiv:2106.07494 [math.CO], 2021.
%H M. A. Deryagina and A. D. Mednykh, <a href="http://dx.doi.org/10.1134/S0037446613040058">On the enumeration of circular maps with given number of edges</a>, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
%H Stoyan Dimitrov, <a href="https://arxiv.org/abs/2103.04332">Sorting by shuffling methods and a queue</a>, arXiv:2103.04332 [math.CO], 2021.
%H J. D. Dixon, <a href="http://dx.doi.org/10.1007/BF01110210">The probability of generating the symmetric group</a>, Math. Z. 110 (1969) 199-205.
%H John D. Dixon, <a href="http://www.combinatorics.org/Volume_12/Abstracts/v12i1r56.html">Asymptotics of Generating the Symmetric and Alternating Groups</a>, Electronic Journal of Combinatorics, vol 11(2), R56.
%H G. Duchamp, F. Hivert, and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0105065">Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras</a>, arXiv:math/0105065 [math.CO], 2001.
%H G. A. Edgar, <a href="http://arxiv.org/abs/0801.4877">Transseries for beginners</a>, arXiv:0801.4877 [math.RA], 2008-2009.
%H Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/2310.06288">Catalan-Spitzer permutations</a>, arXiv:2310.06288 [math.CO], 2023. See p. 19.
%H Jesse Elliott, <a href="https://arxiv.org/abs/1809.06633">Asymptotic expansions of the prime counting function</a>, arXiv:1809.06633 [math.NT], 2018.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 90.
%H A. L. L. Gao, S. Kitaev, and P. B. Zhang. <a href="https://arxiv.org/abs/1605.05490">On pattern avoiding indecomposable permutations</a>, arXiv:1605.05490 [math.CO], 2016.
%H I. M. Gessel and R. P. Stanley <a href="http://people.brandeis.edu/~gessel/homepage/papers/enum.pdf">Algebraic Enumeration</a> (See pages 7-8 for generating function.)
%H R. K. Guy, <a href="/A003320/a003320.pdf">Letter to N. J. A. Sloane, Mar 1974</a>
%H P. Hegarty and A. Martinsson, <a href="http://arxiv.org/abs/1210.4798">On the existence of accessible paths in various models of fitness landscapes</a>, arXiv preprint arXiv:1210.4798 [math.PR], 2012. - From _N. J. A. Sloane_, Jan 01 2013
%H V. Jelínek and P. Valtr, <a href="http://arxiv.org/abs/1307.0027">Splittings and Ramsey Properties of Permutation Classes</a>, arXiv preprint arXiv:1307.0027 [math.CO], 2013.
%H B. R. Jones, <a href="http://summit.sfu.ca/item/14554">On tree hook length formulas, Feynman rules and B-series</a>, Master's thesis, Simon Fraser University, 2014.
%H A. King, <a href="http://dx.doi.org/10.1016/j.disc.2006.01.005">Generating indecomposable permutations</a>, Discrete Math., 306 (2006), 508-518.
%H Y. Koh and S. Ree, <a href="http://dx.doi.org/10.1016/j.disc.2006.11.014">Connected permutation graphs</a>, Discrete Math. 307 (2007), 2628-2635.
%H M. K. Krotter, I. C. Christov, J. M. Ottino and R. M. Lueptow, <a href="http://arxiv.org/abs/1208.2052">Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations</a>, arXiv preprint arXiv:1208.2052 [physics.flu-dyn], 2012. - From _N. J. A. Sloane_, Dec 25 2012
%H Ajay Kumar and Chanchal Kumar, <a href="https://www.ias.ac.in/public/Volumes/pmsc/forthcoming/PMSC-D-17-00160.pdf">Monomial ideals induced by permutations avoiding patterns</a>, 2018.
%H Chanchal Kumar and Amit Roy, <a href="https://arxiv.org/abs/2003.10098">Integer Sequences and Monomial Ideals</a>, arXiv:2003.10098 [math.CO], 2020.
%H Steven Linton, James Propp, Tom Roby, and Julian West, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Roby/roby4.html">Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
%H R. J. Martin and M. J. Kearney, <a href="http://arXiv.org/abs/1103.4936">An exactly solvable self-convolutive recurrence</a>, arXiv:1103.4936 [math.CO], 2011.
%H R. J. Martin and M. J. Kearney, <a href="http://dx.doi.org/10.1007/s00010-010-0051-0">An exactly solvable self-convolutive recurrence</a>, Aequat. Math., 80 (2010), 291-318. see p. 292.
%H Richard J. Martin and Michael J. Kearney, <a href="http://dx.doi.org/10.1007/s00493-014-3183-3">Integral representation of certain combinatorial recurrences</a>, Combinatorica: 35:3 (2015), 309-315.
%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a> (2008); arXiv:0806.3682 [math.CO]. Discrete Math. 310 (2010), no. 24, 3584-3606.
%H P. Ossona de Mendez and P. Rosenstiehl, <a href="http://dx.doi.org/10.1007/s00493-004-0029-4">Transitivity and connectivity of permutations</a>, Combinatorica, 24 (No. 3, 2004), 487-501.
%H L. Panaitopol, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-1-055.pdf">A formula for pi(x) applied to a result of Koninck-Ivić</a>, Nieuw Arch. Wisk. 5/1 55-56 (2000).
%H Vincent Pilaud, <a href="http://arxiv.org/abs/1505.07665">Brick polytopes, lattice quotients, and Hopf algebras</a>, arXiv:1505.07665 [math.CO], 2015.
%H S. Poirier and C. Reutenauer, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-1/PDF/079-090.pdf">Algèbres Hopf de tableaux</a>, Ann. Sci. Math. Quebec 19 (95), no. 1, 79-90.
%H N. Reading <a href="http://dx.doi.org/10.1007/s10801-010-0255-3">Noncrossing partitions and the shard intersection order</a>. J. Algebraic Combin. 33 (2011), no. 4. arXiv preprint <a href="http://arxiv.org/abs/0909.3288">arXiv:0909.3288</a>, [math.CO], 2009.
%H Herman P. Robinson, <a href="/A003116/a003116_1.pdf">Letter to N. J. A. Sloane, Nov 19 1973</a>.
%H R. P. Stanley, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Stanley/stanley80.html">The Descent Set and Connectivity Set of a Permutation</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.8.
%H Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.3.
%H Marcel Wienöbst, Max Bannach, and Maciej Liśkiewicz, <a href="https://arxiv.org/abs/2205.02654">Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications</a>, arXiv:2205.02654 [cs.LG], 2022.
%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 7.
%F G.f.: 2 - 1/Sum_{k>=0} k!*x^k.
%F Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k) [Bowen, 1976].
%F Also coefficients in the divergent series expansion log Sum_{n>=0} n!*x^n = Sum_{n>=1} a(n+1)*x^n/n [Bowen, 1976].
%F a(n) = (-1)^(n-1) * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}.
%F INVERTi transform of factorial numbers, A000142 starting from n=1. - _Antti Karttunen_, May 30 2003
%F Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle A089949. - _Philippe Deléham_, Dec 30 2003
%F a(n+1) = Sum_{k=0..n} A089949(n,k). - _Philippe Deléham_, Oct 16 2006
%F L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ). - _Paul D. Hanna_, Sep 19 2007
%F G.f.: 1+x/(1-x/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-...)))))))) (continued fraction). - _Paul Barry_, Oct 07 2008
%F a(n) = -Sum_{i=0..n} (-1)^i*A090238(n, i) for n > 0. - _Peter Luschny_, Mar 13 2009
%F From _Gary W. Adamson_, Jul 14 2011: (Start)
%F a(n) = upper left term in M^(n-1), M = triangle A128175 as an infinite square production matrix (deleting the first "1"); as follows:
%F 1, 1, 0, 0, 0, 0, ...
%F 2, 2, 1, 0, 0, 0, ...
%F 4, 4, 3, 1, 0, 0, ...
%F 8, 8, 7, 4, 1, 0, ...
%F 16, 16, 15, 11, 5, 1, ...
%F ... (End)
%F O.g.f. satisfies: A(x) = x - x*A(x) + A(x)^2 + x^2*A'(x). - _Paul D. Hanna_, Jul 30 2011
%F From _Sergei N. Gladkovskii_, Jun 24 2012: (Start)
%F Let A(x) be the g.f.; then
%F A(x) = 1/Q(0), where Q(k) = x + 1 + x*k - (k+2)*x/Q(k+1).
%F A(x) = (1-1/U(0))/x, when U(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/U(k+1))). (End)
%F From _Sergei N. Gladkovskii_, May, Jun, Aug 2013: (Start)
%F Continued fractions:
%F G.f.: 1 - G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))).
%F G.f.: (x/2)*G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1/2) + 1/G(k+1))).
%F G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1)).
%F G.f.: 1 - 1/G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1)))).
%F G.f.: x*W(0), where W(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+2)/(x*(k+2) - 1/W(k+1)))).
%F (End)
%F a(n) = A233824(n-1) if n > 0. (Proof. Set b(n) = A233824(n), so that b(n) = n*n! - Sum_{k=1..n-1} k!*b(n-k). To get a(n+1) = b(n) for n >= 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.) - _Jonathan Sondow_, Dec 19 2013
%F a(n) ~ n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10), for coefficients see A260503. - _Vaclav Kotesovec_, Jul 27 2015
%F For n>0, a(n) = (A059439(n) - A259472(n))/2. - _Vaclav Kotesovec_, Aug 03 2015
%F From _Peter Bala_, May 23 2017: (Start)
%F G.f.: 1 + x/(1 + x - 2*x/(1 + 2*x - 3*x/(1 + 3*x - 4*x/(1 + 4*x - ...)))). Cf. A000698.
%F G.f.: 1/(1 - x/(1 + x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ...))))))))). (End)
%F Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - _Mikhail Kurkov_, Apr 26 2024
%e G.f. = 1 + x + x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 461*x^6 + 3447*x^7 + 29093*x^8 + ...
%e From _Peter Luschny_, Aug 03 2022: (Start)
%e A permutation p in [n] (where n >= 0) is reducible if there exists an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation at i.
%e Examples:
%e * () is not reducible since there is no index i which splits (). (=> a(0) = 1)
%e * (1) is not reducible since there is no index i which splits (1). (=> a(1) = 1)
%e * (1, 2) is reducible since index 1 splits (1, 2) as p(1) < p(2).
%e * (2, 1) is not reducible since at the only potential splitting point i = 1 we have p(1) > p(2). (=> a(2) = 1)
%e * For n = 3 we have (1, 2, 3), (1, 3, 2), and (2, 1, 3) are reducible and (2, 3, 1), (3, 1, 2), and (3, 2, 1) are irreducible. (End)
%p INVERTi([seq(n!,n=1..20)]);
%p A003319 := proc(n) option remember; n! - add((n-j)!*A003319(j), j=1..n-1) end;
%p [seq(A003319(n), n=0..50)]; # _N. J. A. Sloane_, Dec 28 2011
%p series(2 - 1/hypergeom([1,1], [], x), x=0,50); # _Mark van Hoeij_, Apr 18 2013
%t a[n_] := a[n] = n! - Sum[k!*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 11 2011, after given formula *)
%t CoefficientList[Assuming[Element[x,Reals],Series[2-E^(1/x)* x/ExpIntegralEi[1/x],{x,0,20}]],x] (* _Vaclav Kotesovec_, Mar 07 2014 *)
%t a[ n_] := If[ n < 2, 1, a[n] = (n - 2) a[n - 1] + Sum[ a[k] a[n - k], {k, n - 1}]]; (* _Michael Somos_, Feb 23 2015 *)
%t Table[SeriesCoefficient[1 + x/(1 + ContinuedFractionK[-Floor[(k + 2)/2]*x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 29 2017 *)
%o (PARI) {a(n) = my(A); if( n<1, 1, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (k - 2) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 24 2011 */
%o (PARI) {if(n<1,1,a(n)=local(A=x);for(i=1,n,A=x-x*A+A^2+x^2*A' +x*O(x^n));polcoeff(A,n))} /* _Paul D. Hanna_, Jul 30 2011 */
%o (Sage)
%o def A003319_list(len):
%o R, C = [1], [1] + [0] * (len - 1)
%o for n in range(1, len):
%o for k in range(n, 0, -1):
%o C[k] = C[k - 1] * k
%o C[0] = -sum(C[k] for k in range(1, n + 1))
%o R.append(-C[0])
%o return R
%o print(A003319_list(21)) # _Peter Luschny_, Feb 19 2016
%Y See A167894 for another version.
%Y Bisections give A272656, A272657.
%Y Row sums of A111184 and A089949.
%Y Leading diagonal of A059438. A diagonal of A263484.
%Y Cf. A051296, A084938, A074664, A113869, A144107.
%Y Cf. A260503, A259472, A261214, A261239, A261253, A261254.
%Y Cf. A090238, A000698, A356291 (reducible permutations).
%Y Column k=0 of A370380 and A370381 (without pair of initial terms and with different offset).
%K nonn,easy,nice,changed
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_, Jan 26 2000
%E Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002
%E Added a(0)=0 (some of the formulas may now need adjusting). - _N. J. A. Sloane_, Sep 12 2012
%E Edited and set a(0) = 1 by _Peter Luschny_, Aug 03 2022
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