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%I M1913 N0755 #268 Apr 19 2024 09:23:30
%S 1,-1,0,1,1,-2,-9,-9,50,267,413,-2180,-17731,-50533,110176,1966797,
%T 9938669,8638718,-278475061,-2540956509,-9816860358,27172288399,
%U 725503033401,5592543175252,15823587507881,-168392610536153,-2848115497132448,-20819319685262839
%N Rao Uppuluri-Carpenter numbers (or complementary Bell numbers): e.g.f. = exp(1 - exp(x)).
%C Alternating row sums of Stirling2 triangle A048993.
%C Related to the matrix-exponential of the Pascal-matrix, see A000110 and A011971. - _Gottfried Helms_, Apr 08 2007
%C Row sums of triangle A144185 = a signed, shifted version of A000587: (1, 0, 1, 1, 2, 9, -9, 50, -267, -413, -2180, ...). Triangle A144185 is generated from A118433, the self-inverse triangle. - _Gary W. Adamson_, Sep 13 2008
%C Closely linked to A000110 and especially the contribution there of Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007, by offering what is a complementary finding.
%C Number of set partitions of 1..n with an even number of parts, minus the number of such partitions with an odd number of parts. - _Franklin T. Adams-Watters_, May 04 2010
%C After -2, the smallest prime is a(36) = -1454252568471818731501051, no others through a(100). What is the first prime >0 in the sequence? - _Jonathan Vos Post_, Feb 02 2011
%C a(723) ~ 1.9*10^1265 is almost certainly prime. - _D. S. McNeil_, Feb 02 2011
%C Stirling transform of a(n) = [1, -1, 0, 1, 1, ...] is A033999(n) = [1, -1, 1, -1, 1, ...]. - _Michael Somos_, Mar 28 2012
%C Negated coefficients in the asymptotic expansion: A005165(n)/n! ~ 1 - 1/n + 1/n^2 + 0/n^3 - 1/n^4 - 1/n^5 + 2/n^6 + 9/n^7 + 9/n^8 - 50/n^9 - 267/n^10 - 413/n^11 + O(1/n^12), starting from the O(1/n) term. - _Vladimir Reshetnikov_, Nov 09 2016
%C Named after Venkata Ramamohana Rao Uppuluri and John A. Carpenter of the Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee. They are called "Rényi numbers" by Fekete (1999), after the Hungarian mathematician Alfréd Rényi (1921-1970). - _Amiram Eldar_, Mar 11 2022
%D N. A. Kolokolnikova, Relations between sums of certain special numbers (Russian), in Asymptotic and enumeration problems of combinatorial analysis, pp. 117-124, Krasnojarsk. Gos. Univ., Krasnoyarsk, 1976.
%D Alfréd Rényi, Új modszerek es eredmenyek a kombinatorikus analfzisben. I. MTA III Oszt. Ivozl., Vol. 16 (1966), pp. 7-105.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D M. V. Subbarao and A. Verma, Some remarks on a product expansion. An unexplored partition function, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FL, 1999), pp. 267-283, Kluwer, Dordrecht, 2001.
%H Alois P. Heinz, <a href="/A000587/b000587.txt">Table of n, a(n) for n = 0..595</a> (first 101 terms from T. D. Noe)
%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. See Example 31. - _N. J. A. Sloane_, May 24 2014
%H W. Asakly, A. Blecher, C. Brennan, A. Knopfmacher, T. Mansour, and S. Wagner, <a href="https://doi.org/10.1016/j.jmaa.2014.02.061">Set partition asymptotics and a conjecture of Gould and Quaintance</a>, Journal of Mathematical Analysis and Applications, Volume 416, Issue 2 (15 August 2014), Pages 672-682.
%H Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, <a href="http://www.tulane.edu/~vhm/papers_html/final-bell.pdf">Complementary Bell numbers: arithmetical properties and Wilf's conjecture</a>.
%H S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero2/barbero7.html">A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences </a>, J. Int. Seq., Vol. 13 (2010), Article 10.9.7.
%H R. E. Beard, <a href="/A000587/a000587.pdf">On the Coefficients in the Expansion of e^(e^t) and e^(-e^t)</a>, J. Institute of Actuaries, Vol. 76 (1950), pp. 152-163. [Annotated scanned copy]
%H Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, <a href="http://arxiv.org/abs/1505.03474">State complexity of catenation combined with a boolean operation: a unified approach</a>, arXiv:1505.03474 [cs.FL], 2015.
%H Valerio De Angelis and Dominic Marcello, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.123.6.557">Wilf's Conjecture</a>, The American Mathematical Monthly, Vol. 123, No. 6 (2016), pp. 557-573.
%H S. de Wannemacker, T. Laffey and R. Osburn, <a href="https://arxiv.org/abs/math/0608085">On a conjecture of Wilf</a>, arXiv:math/0608085 [math.NT], 2006-2007.
%H Branko Dragovich, <a href="https://arxiv.org/abs/1702.02569">On Summation of p-Adic Series</a>, arXiv:1702.02569 [math.NT], 2017.
%H Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, <a href="http://arxiv.org/abs/1508.05079">Summation of p-Adic Functional Series in Integer Points</a>, arXiv:1508.05079, 2015
%H B. Dragovich and N. Z. Misic, <a href="http://dx.doi.org/10.1134/S2070046614040025">p-Adic invariant summation of some p-adic functional series, P-Adic Numbers</a>, Ultrametric Analysis, and Applications, Volume 6, Issue 4 (October 2014), pp. 275-283.
%H Antal E. Fekete, <a href="https://cms.math.ca/publications/crux/issue/?volume=25&issue=5">Apropos Bell and Stirling Numbers</a>, Crux Mathematicorum, Vol. 25, No. 5 (1999), pp. 274-281.
%H B. Harris and L. Schoenfeld, <a href="https://doi.org/10.1215/ijm/1256054216">Asymptotic expansions for the coefficients of analytic functions</a>, Ill. J. Math., Vol. 12 (1968), pp. 264-277.
%H M. Klazar, <a href="https://www.jstor.org/stable/3647908">Counting even and odd partitions</a>, Amer. Math. Monthly, Vol. 110, No. 6 (2003), pp. 527-532.
%H M. Klazar, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00014-1">Bell numbers, their relatives and algebraic differential equations</a>, J. Combin. Theory, A 102 (2003), 63-87.
%H A. Knopfmacher and M. E. Mays, <a href="http://math.colgate.edu/~integers/b4/b4.pdf">Graph compositions I: Basic enumerations</a>, Integers, Vol. 1 (2001), Article A4. (See the first two columns of the table on p. 9.)
%H Vaclav Kotesovec, <a href="/A000587/a000587.jpg">Plot of |a(n)/n!|^(1/n) / |exp(1/W(-n))/W(-n)| for n = 1..40000</a>, where W is the LambertW function.
%H Peter J. Larcombe, Jack Sutton, and James Stanton, <a href="https://pjm.ppu.edu/sites/default/files/papers/PJM_12%282%29_2023_609_to_619.pdf">A note on the constant 1/e</a>, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619. See p. 617.
%H J. W. Layman and C. L. Prather, <a href="https://doi.org/10.1016/0022-247X(83)90025-2">Generalized Bell numbers and zeros of successive derivatives of an entire function</a>, Journal of Mathematical Analysis and Applications, Volume 96, Issue 1 (15 October 1983), Pages 42-51.
%H Toufik Mansour and Mark Shattuck, <a href="https://doi.org/10.2298/AADM210223009M">Counting subword patterns in permutations arising as flattened partitions of sets</a>, Appl. Anal. Disc. Math. (2022), OnLine-First (00):9-9.
%H T. Mansour, M. Shattuck and D. G. L. Wang, <a href="http://arxiv.org/abs/1306.3355">Recurrence relations for patterns of type (2, 1) in flattened permutations</a>, arXiv preprint arXiv:1306.3355 [math.CO], 2013.
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterIII/page5.htm">Notebook entry</a>.
%H V. R. Rao Uppuluri and J. A. Carpenter, <a href="http://www.fq.math.ca/Scanned/7-4/uppuluri.pdf">Numbers generated by the function exp(1-e^x)</a>, Fib. Quart., Vol. 7, No. 4 (1969), pp. 437-448.
%H Frank Ruskey and Jennifer Woodcock, <a href="https://doi.org/10.1007/978-3-642-25011-8_23">The Rand and block distances of pairs of set partitions</a>, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
%H Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, <a href="https://doi.org/10.1016/j.jda.2012.04.003">Counting and computing the Rand and block distances of pairs of set partitions</a>, Journal of Discrete Algorithms, Volume 16 (October 2012), Pages 236-248. [_N. J. A. Sloane_, Oct 03 2012]
%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq., Vol. 14 (2011), Article 11.9.7.
%H D. Subedi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Subedi/subedi3.html">Complementary Bell Numbers and p-adic Series</a>, J. Int. Seq., Vol. 17 (2014), Article 14.3.1.
%H A. Vieru, <a href="http://arxiv.org/abs/1107.2938">Agoh's conjecture: its proof, its generalizations, its analogues</a>, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ComplementaryBellNumber.html">Complementary Bell Number</a>.
%H D. Wuilquin, <a href="/A000587/a000587_1.pdf">Letters to N. J. A. Sloane, August 1984</a>.
%H Yifan Yang, <a href="https://doi.org/10.37236/1563">On a multiplicative partition function</a>, Electron. J. Combin., Vol. 8, No. 1 (2001), Research Paper 19.
%F a(n) = e*Sum_{k>=0} (-1)^k*k^n/k!. - _Benoit Cloitre_, Jan 28 2003
%F E.g.f.: exp(1 - e^x).
%F a(n) = Sum_{k=0..n} (-1)^k S2(n, k), where S2(i, j) are the Stirling numbers of second kind A008277.
%F G.f.: (x/(1-x))*A(x/(1-x)) = 1 - A(x); the binomial transform equals the negative of the sequence shifted one place left. - _Paul D. Hanna_, Dec 08 2003
%F With different signs: g.f.: Sum_{k>=0} x^k/Product_{L=1..k} (1 + L*x).
%F Recurrence: a(n) = -Sum_{i=0..n-1} a(i)*C(n-1, i). - _Ralf Stephan_, Feb 24 2005
%F Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential); then a(n) = PE^-1 [n,1]. - _Gottfried Helms_, Apr 08 2007
%F Take the series 0^n/0! - 1^n/1! + 2^n/2! - 3^n/3! + 4^n/4! + ... If n=0 then the result will be 1/e, where e = 2.718281828... If n=1, the result will be -1/e. If n=2, the result will be 0 (i.e., 0/e). As we continue for higher natural number values of n sequence for the Roa Uppuluri-Carpenter numbers is generated in the numerator, i.e., 1/e, -1/e, 0/e, 1/e, 1/e, -2/e, -9/e, -9/e, 50/e, 267/e, ... . - Peter Collins (pcolins(AT)eircom.net), Jun 04 2007
%F The sequence (-1)^n*a(n), with general term Sum_{k=0..n} (-1)^(n-k)*S2(n, k), has e.g.f. exp(1-exp(-x)). It also has Hankel transform (-1)^C(n+1,2)*A000178(n) and binomial transform A109747. - _Paul Barry_, Mar 31 2008
%F G.f.: 1 / (1 + x / (1 - x / (1 + x / (1 - 2*x / (1 + x / (1 - 3*x / (1 + x / ...))))))). - _Michael Somos_, May 12 2012
%F From _Sergei N. Gladkovskii_, Sep 28 2012 to Feb 07 2014: (Start)
%F Continued fractions:
%F G.f.: -1/U(0) where U(k) = x*k - 1 - x + x^2*(k+1)/U(k+1).
%F G.f.: 1/(U(0)+x) where U(k) = 1 + x - x*(k+1)/(1 + x/U(k+1)).
%F G.f.: 1+x/G(0) where G(k) = x*k - 1 + x^2*(k+1)/G(k+1).
%F G.f.: (1 - G(0))/(x+1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x+1/G(k+1) )).
%F G.f.: 1 + x/(G(0)-x) where G(k) = x*k + 2*x - 1 - x*(x*k+x-1)/G(k+1).
%F G.f.: G(0)/(1+x), where G(k) = 1-x^2*(k+1)/(x^2*(k+1)+(x*k-1-x)*(x*k-1)/G(k+1)).
%F (End)
%F a(n) = B_n(-1), where B_n(x) is n-th Bell polynomial. - _Vladimir Reshetnikov_, Oct 20 2015
%F From _Mélika Tebni_, May 20 2022: (Start)
%F a(n) = Sum_{k=0..n} (-1)^k*Bell(k)*A129062(n, k).
%F a(n) = Sum_{k=0..n} (-1)^k*k!*A130191(n, k). (End)
%e G.f. = 1 - x + x^3 + x^4 - 2*x^5 - 9*x^6 - 9*x^7 + 50*x^8 + 267*x^9 + 413*x^10 - ...
%p b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
%p add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 28 2016
%t Table[ -1 * Sum[ (-1)^( k + 1) StirlingS2[ n, k ], {k, 0, n} ], {n, 0, 40} ]
%t With[{nn=30},CoefficientList[Series[Exp[1-Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 04 2011 *)
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ 1 - Exp[x]], {x, 0, n}]]; (* _Michael Somos_, May 27 2014 *)
%t a[ n_] := If[ n < 0, 0, With[{m = n + 1}, SeriesCoefficient[ Series[ Nest[ x Factor[ 1 - # /. x -> x / (1 - x)] &, 0, m], {x, 0, m}], {x, 0, m}]]]; (* _Michael Somos_, May 27 2014 *)
%t Table[BellB[n, -1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)
%t b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k -= b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k; k*(-1)^n, {n, 1, 40}]}] (* _Vaclav Kotesovec_, Sep 09 2019 *)
%o (Sage) expnums(26, -1) # _Zerinvary Lajos_, May 15 2009
%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - exp( x + x * O(x^n))), n))}; /* _Michael Somos_, Mar 14 2011 */
%o (PARI) {a(n) = local(A); if( n<0, 0, n++; A = O(x); for( k=1, n, A = x - x * subst(A, x, x / (1 - x))); polcoeff( A, n))}; /* _Michael Somos_, Mar 14 2011 */
%o (PARI) Vec(serlaplace(exp(1 - exp(x+O(x^99))))) /* _Joerg Arndt_, Apr 01 2011 */
%o (PARI) a(n)=round(exp(1)*suminf(k=0,(-1)^k*k^n/k!))
%o vector(20,n,a(n-1)) \\ _Derek Orr_, Sep 19 2014 -- a direct approach
%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(1 - exp(x)))) \\ _Michel Marcus_, Sep 19 2014
%o (Python) # The objective of this implementation is efficiency.
%o # n -> [a(0), a(1), ..., a(n)] for n > 0.
%o def A000587_list(n):
%o A = [0 for i in range(n)]
%o A[n-1] = 1
%o R = [1]
%o for j in range(0, n):
%o A[n-1-j] = -A[n-1]
%o for k in range(n-j, n):
%o A[k] += A[k-1]
%o R.append(A[n-1])
%o return R
%o # _Peter Luschny_, Apr 18 2011
%o (Python)
%o # Python 3.2 or higher required
%o from itertools import accumulate
%o A000587, blist, b = [1,-1], [1], -1
%o for _ in range(30):
%o blist = list(accumulate([b]+blist))
%o b = -blist[-1]
%o A000587.append(b) # _Chai Wah Wu_, Sep 19 2014
%o (Haskell)
%o a000587 n = a000587_list !! n
%o a000587_list = 1 : f a007318_tabl [1] where
%o f (bs:bss) xs = y : f bss (y : xs) where y = - sum (zipWith (*) xs bs)
%o -- _Reinhard Zumkeller_, Mar 04 2014
%Y Cf. A000110, A011971 (base triangle PE), A078937 (PE^2).
%Y Cf. A144185, A118433, A007318, A153229, A213170.
%K sign,easy,nice
%O 0,6
%A _N. J. A. Sloane_
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