|
%I #92 Apr 17 2021 03:40:21
%S 1,1,3,14,88,694,6578,72792,920904,13109088,207360912,3608233056,
%T 68495486640,1408631978064,31197601660080,740303842925184,
%U 18738231641600256,503937595069600896,14349899305396086912,431322634732516137216,13646841876634025159424
%N Number of factorizations of permutations of n letters into ordered cycles.
%C a(n) is the number of ways to seat n people at an unspecified number of circular tables and then linearly order the nonempty tables. - _Geoffrey Critzer_, Mar 18 2009
%C The terms of this sequence for n >= 1 are the row sums of A008275^2, the unsigned version of A039814. - _Peter Bala_, Jul 22 2014
%C Signed sequence is the base for an Appell sequence of polynomials with the e.g.f. e^(x*t)/[log(1+t) + 1] = exp(P(.,x),t) that is the umbral compositional inverse for A238385, reverse of A111492, i.e., umbrally evaluated UP(n,P(.,t))= x^n = P(n,UP(.,t)) where UP(n,t) are the polynomials of A238385. Umbrally evaluated means letting (A(.,t))^n = A(n,t) after substituting A for the independent variable of the polynomial. - _Tom Copeland_, Nov 15 2014
%C a(n) is the number of unimodal rooted forests on n labeled nodes (i.e., those forests that avoid the patterns 213 and 312). - _Kassie Archer_, Aug 30 2018
%C Number of permutations of [n] where fixed points at index j are j-colored and all other points are unicolored. - _Alois P. Heinz_, Apr 24 2020
%H Alois P. Heinz, <a href="/A007840/b007840.txt">Table of n, a(n) for n = 0..420</a>
%H K. Anders and K. Archer, <a href="https://arxiv.org/abs/1607.03046">Rooted forests that avoid sets of permutations</a>, arXiv:1607.03046 [math.CO], 2016-2017.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 119.
%H W. S. Gray and M. Thitsa, <a href="http://dx.doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=122">Encyclopedia of Combinatorial Structures 122</a>
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H A. Knopfmacher and J. N. Ridley, <a href="http://dx.doi.org/10.1137/0406031">Reciprocal sums over partitions and compositions</a>, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
%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.
%F a(n) = Sum_{k=1..n} k! * s(n, k), s(n, k) = unsigned Stirling number of first kind; E.g.f. 1/(1+log(1-z)).
%F For n>0, a(n) is the permanent of the n X n matrix with entries a(i, i) = i and a(i, j) = 1 elsewhere. - _Philippe Deléham_, Dec 09 2003
%F a(n) = A052860(n)/n for n >= 1.
%F a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n >= 1 with a(0)=1. - _Paul D. Hanna_, Jul 19 2006
%F E.g.f.: B(A(x)) where B(x) = 1/(1-x) and A(x) = log(1/(1-x)). - _Geoffrey Critzer_, Mar 18 2009
%F a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - _Peter Bala_, Nov 25 2011
%F E.g.f.: 1/(1+log(1-x)) = 1/(1 - x/(1 - x/(2 - x/(3 - 4*x/(4 - 4*x/(5 - 9*x/(6 - 9*x/(7 - 16*x/(8 - 16*x/(9 - ...)))))))))), a continued fraction. - _Paul D. Hanna_, Dec 31 2011
%F a(n) ~ n! * exp(n)/(exp(1)-1)^(n+1). - _Vaclav Kotesovec_, Jun 21 2013
%p a:= proc(n) a(n):= n!*`if`(n=0, 1, add(a(k)/(k!*(n-k)), k=0..n-1)) end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 06 2012
%t Table[Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 20}] (* _Geoffrey Critzer_, Mar 18 2009 *)
%o (PARI) a(n)=n!*polcoeff(1/(1+log(1-x +x*O(x^n))),n) /* _Paul D. Hanna_, Jul 19 2006 */
%o (PARI) {a(n)=local(CF=1+x*O(x)); for(k=0, n-1, CF=1/((n-k)-((n-k+1)\2)^2*x*CF)); n!*polcoeff(1/(1-x*CF), n)} /* _Paul D. Hanna_, Jul 19 2006 */
%o (Sage)
%o def A007840_list(len):
%o f, R, C = 1, [1], [1]+[0]*len
%o for n in (1..len):
%o f *= n
%o for k in range(n, 0, -1):
%o C[k] = -C[k-1]*((k-1)/k if k>1 else 1)
%o C[0] = sum((-1)^k*C[k] for k in (1..n))
%o R.append(C[0]*f)
%o return R
%o print(A007840_list(20)) # _Peter Luschny_, Feb 21 2016
%Y Cf. A052860. Row sums of unsigned version of A039814.
%Y Cf. A238385, A111492.
%K nonn
%O 0,3
%A _Arnold Knopfmacher_
%E Extended June 1995
| 