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

0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017, 29272410, 410815351, 6231230412, 101560835377, 1769925341366, 32838929702671, 646218442877520, 13441862819232001, 294656673023216946, 6788407001443004647, 163962850573039534580, 4142654439686285737201

Offset: 0

Author

Vladeta Jovovic, Nov 05 2003

Keywords

• nonn
• easy

Comments

From Peter Bala, Mar 27 2022: (Start)

a(2*n) is even; in fact, 2*n*(2*n-1)*(2n-2) divides a(2*n). a(2*n+1) is odd.

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)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..444
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
• N. J. A. Sloane, LAH transform

Crossrefs

Cf. A000262, A001710, A027187, A024429, A024430, A088312, A109777, A111884.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); [0] cat Coefficients(R!(Laplace( Sinh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022

Maple

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-j, 1-t)*binomial(n-1, j-1)*j!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
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

Mathematica

With[{m=30}, CoefficientList[Series[Sinh[x/(1-x)], {x,0,m}], x] * Range[0,m]!] (* Vaclav Kotesovec, Jul 04 2015 *)

PARI

my(x='x+O('x^66)); concat(0, Vec(serlaplace(sinh(x/(1-x))))) \\ Joerg Arndt, Jul 16 2013

SageMath

def A088313_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x/(1-x)) ).egf_to_ogf().list()
A088313_list(40) # G. C. Greubel, Dec 13 2022

Formula

E.g.f.: sinh(x/(1-x)).

a(n) = Sum_{k=1..floor((n+1)/2)} n!/(2*k-1)!*binomial(n-1, 2*k-2).

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

a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015

a(n) = (1/2)*(A000262(n) - (-1)^n*A111884(n)). - Peter Bala, Mar 27 2022

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

Extensions

a(0)=0 prepended by Alois P. Heinz, May 10 2016