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A088312 Number of sets of lists (cf. A000262) with even number of lists.

%I A088312 #36 Dec 14 2022 06:30:08
%S A088312 1,0,1,6,37,260,2101,19362,201097,2326536,29668681,413257790,
%T A088312 6238931821,101415565836,1765092183037,32734873484250,644215775792401,
%U A088312 13404753632014352,293976795292186897,6775966692145553526,163735077313046119861,4138498600079573989140
%N A088312 Number of sets of lists (cf. A000262) with even number of lists.
%C A088312 From _Peter Bala_, Mar 27 2022: (Start)
%C A088312 a(2*n) is odd ; a(2*n+1) is even.
%C A088312 If k is odd then k*(k-1) divides a(k). Consequently, 6 divides a(6*n+3), 10 divides a(10*n+5), 14 divides a(14*n+7), and in general, if k is odd then 2*k divides a(2*k*n + k).
%C A088312 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 A088312 Alois P. Heinz, <a href="/A088312/b088312.txt">Table of n, a(n) for n = 0..444</a>
%H A088312 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%H A088312 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 A088312 N. J. A. Sloane, <a href="/transforms.txt">LAH transform</a>
%F A088312 E.g.f.: cosh(x/(1-x)).
%F A088312 a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
%F A088312 a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - _Vaclav Kotesovec_, Jul 04 2015
%F A088312 a(n+4) - 2*(2*n+5)*a(n+3) + (6*n^2+24*n+23)*a(n+2) - 2*(n+1)*(n+2)*(2*n+3)*a(n+1) + n*(n+1)^2*(n+2)*a(n) = 0. - _Emanuele Munarini_, Sep 03 2017
%F A088312 a(n) = (1/2)*(A000262(n) + (-1)^n*A111884(n)). - _Peter Bala_, Mar 27 2022
%F A088312 a(n) = (1/2)*(n-1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4) for n >= 1. - _Peter Luschny_, Dec 14 2022
%p A088312 b:= proc(n, t) option remember; `if`(n=0, t, add(
%p A088312       b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
%p A088312     end:
%p A088312 a:= n-> b(n, 1):
%p A088312 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016
%p A088312 A088312 := n -> ifelse(n=0, 1, (1/2)*(n - 1)*n!*hypergeom([1 - n/2, 3/2 - n/2], [3/2, 3/2, 2], 1/4)): seq(simplify(A088312(n)), n = 0..21); # _Peter Luschny_, Dec 14 2022
%t A088312 With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* _Vaclav Kotesovec_, Jul 04 2015 *)
%t A088312 Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* _Emanuele Munarini_, Sep 03 2017 *)
%o A088312 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(x/(1-x)) ))); // _G. C. Greubel_, Dec 13 2022
%o A088312 (SageMath)
%o A088312 def A088312_list(prec):
%o A088312     P.<x> = PowerSeriesRing(QQ, prec)
%o A088312     return P( cosh(x/(1-x)) ).egf_to_ogf().list()
%o A088312 A088312_list(40) # _G. C. Greubel_, Dec 13 2022
%Y A088312 Cf. A000262, A001710, A027187, A024429, A024430, A088313, A111884.
%K A088312 nonn,easy
%O A088312 0,4
%A A088312 _Vladeta Jovovic_, Nov 05 2003
%E A088312 More terms from _Vaclav Kotesovec_, Jul 04 2015
%E A088312 a(0)-a(1) prepended by _Alois P. Heinz_, May 10 2016