OFFSET
0,3
COMMENTS
Also coefficient of x^n in the expansion of 1/(n+1) * (1 + n*x + x^2)^(n+1). - Seiichi Manyama, May 06 2019
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..386
FORMULA
a(n) = Sum_{j=0..floor(n/2)} ((j+1)*n^(n-2*j)*n!)/((j+1)!^2*(n-2*j)!).
a(n) ~ BesselI(1,2) * n^n. - Vaclav Kotesovec, Dec 12 2014
From Ilya Gutkovskiy, Sep 21 2017: (Start)
a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 4)*x^2))/(2*x^2).
a(n) = [x^n] 1/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - ...))))), a continued fraction. (End)
MATHEMATICA
Flatten[{1, Table[n^n*HypergeometricPFQ[{1/2-n/2, -n/2}, {2}, 4/n^2], {n, 1, 20}]}] (* Vaclav Kotesovec, Dec 12 2014 *)
PROG
(Sage)
a = lambda n: 1 if n==0 else n^n*hypergeometric([1/2-n/2, -n/2], [2], 4/n^2).simplify()
[a(n) for n in range(20)]
(PARI) {a(n) = sum(k=0, n\2, n^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1))} \\ Seiichi Manyama, May 05 2019
(PARI) {a(n) = polcoef((1+n*x+x^2)^(n+1)/(n+1), n)} \\ Seiichi Manyama, May 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 12 2014
STATUS
approved