OFFSET
0,4
COMMENTS
From Peter Bala, Mar 27 2022: (Start)
a(2*n) is odd ; a(2*n+1) is even.
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).
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
FORMULA
E.g.f.: cosh(x/(1-x)).
a(n) = Sum_{k=1..floor(n/2)} n!/(2*k)!*binomial(n-1,2*k-1).
a(n) ~ 2^(-3/2) * n^(n-1/4) * exp(2*sqrt(n)-n-1/2). - Vaclav Kotesovec, Jul 04 2015
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
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
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, 1):
seq(a(n), n=0..30); # Alois P. Heinz, May 10 2016
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
MATHEMATICA
With[{m=30}, CoefficientList[Series[Cosh[x/(1-x)], {x, 0, m}], x] * Range[0, m]!] (* Vaclav Kotesovec, Jul 04 2015 *)
Table[Sum[n!/(2*k)! Binomial[n - 1, 2*k - 1], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Emanuele Munarini, Sep 03 2017 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Cosh(x/(1-x)) ))); // G. C. Greubel, Dec 13 2022
(SageMath)
def A088312_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( cosh(x/(1-x)) ).egf_to_ogf().list()
A088312_list(40) # G. C. Greubel, Dec 13 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Nov 05 2003
EXTENSIONS
More terms from Vaclav Kotesovec, Jul 04 2015
a(0)-a(1) prepended by Alois P. Heinz, May 10 2016
STATUS
approved