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A088313
Number of "sets of lists" (cf. A000262) with an odd number of lists.
4
0, 1, 2, 7, 36, 241, 1950, 18271, 193256, 2270017, 29272410, 410815351, 6231230412, 101560835377, 1769925341366, 32838929702671, 646218442877520, 13441862819232001, 294656673023216946, 6788407001443004647, 163962850573039534580, 4142654439686285737201
OFFSET
0,3
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
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
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
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 *)
PROG
(PARI) my(x='x+O('x^66)); concat(0, Vec(serlaplace(sinh(x/(1-x))))) \\ Joerg Arndt, Jul 16 2013
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); [0] cat Coefficients(R!(Laplace( Sinh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022
(SageMath)
def A088313_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( sinh(x/(1-x)) ).egf_to_ogf().list()
A088313_list(40) # G. C. Greubel, Dec 13 2022
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Nov 05 2003
EXTENSIONS
a(0)=0 prepended by Alois P. Heinz, May 10 2016
STATUS
approved