cp's OEIS Frontend

A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.

%I A088313 #33 Dec 14 2022 06:29:58
%S A088313 0,1,2,7,36,241,1950,18271,193256,2270017,29272410,410815351,
%T A088313 6231230412,101560835377,1769925341366,32838929702671,646218442877520,
%U A088313 13441862819232001,294656673023216946,6788407001443004647,163962850573039534580,4142654439686285737201
%N A088313 Number of "sets of lists" (cf. A000262) with an odd number of lists.
%C A088313 From _Peter Bala_, Mar 27 2022: (Start)
%C A088313 a(2*n) is even; in fact, 2*n*(2*n-1)*(2n-2) divides a(2*n). a(2*n+1) is odd.
%C A088313 For a positive integer k, a(n+2*k) - a(n) is divisible by k. Thus the sequence obtained by taking a(n) modulo k is purely periodic with period 2*k. Calculation suggests that when k is even the exact period equals k, and when k is odd the exact period equals 2*k. (End)
%H A088313 Alois P. Heinz, <a href="/A088313/b088313.txt">Table of n, a(n) for n = 0..444</a>
%H A088313 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A088313 N. J. A. Sloane, <a href="/transforms.txt">LAH transform</a>
%F A088313 E.g.f.: sinh(x/(1-x)).
%F A088313 a(n) = Sum_{k=1..floor((n+1)/2)} n!/(2*k-1)!*binomial(n-1, 2*k-2).
%F A088313 E.g.f.: sinh(x/(1-x)) = x/(2-2*x)*E(0), where E(k)= 1 + 1/( 1 - x^2/(x^2 + 2*(1-x)^2*(k+1)*(2*k+3)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%F A088313 a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - _Vaclav Kotesovec_, Jul 04 2015
%F A088313 a(n) = (1/2)*(A000262(n) - (-1)^n*A111884(n)). - _Peter Bala_, Mar 27 2022
%F A088313 a(n) = n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4) for n >= 1. - _Peter Luschny_, Dec 14 2022
%p A088313 b:= proc(n, t) option remember; `if`(n=0, t, add(
%p A088313       b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
%p A088313     end:
%p A088313 a:= n-> b(n, 0):
%p A088313 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016
%p A088313 A088313 := n -> ifelse(n=0, 0, n!*hypergeom([1/2 - n/2, 1 - n/2], [1/2, 1, 3/2], 1/4)): seq(simplify(A088313(n)), n = 0..21); # _Peter Luschny_, Dec 14 2022
%t A088313 With[{m=30}, CoefficientList[Series[Sinh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* _Vaclav Kotesovec_, Jul 04 2015 *)
%o A088313 (PARI) my(x='x+O('x^66)); concat(0, Vec(serlaplace(sinh(x/(1-x))))) \\ _Joerg Arndt_, Jul 16 2013
%o A088313 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); [0] cat Coefficients(R!(Laplace( Sinh(x/(1-x)) ))); // _G. C. Greubel_, Dec 13 2022
%o A088313 (SageMath)
%o A088313 def A088313_list(prec):
%o A088313     P.<x> = PowerSeriesRing(QQ, prec)
%o A088313     return P( sinh(x/(1-x)) ).egf_to_ogf().list()
%o A088313 A088313_list(40) # _G. C. Greubel_, Dec 13 2022
%Y A088313 Cf. A000262, A001710, A027187, A024429, A024430, A088312, A109777, A111884.
%K A088313 nonn,easy
%O A088313 0,3
%A A088313 _Vladeta Jovovic_, Nov 05 2003
%E A088313 a(0)=0 prepended by _Alois P. Heinz_, May 10 2016