A195509 Expansion of e.g.f. (exp(x*exp(x)) + exp(x/exp(x)))/2.
1, 1, 1, 4, 25, 96, 481, 3368, 20721, 141760, 1146721, 9098112, 77652169, 726208640, 6891125697, 69344336896, 738718169569, 8076031881216, 92647353941569, 1106883171037184, 13616813607795321, 174298975125127168, 2304515271134124577
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 96*x^5/5! + 481*x^6/6! +...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..541
Programs
-
Maple
a := proc(n) local m: add(binomial(n, 2*m)*(n - 2*m)^(2*m), m = 0 .. floor(1/2*n - 1/2)): end proc: seq(a(n), n = 1..30); # Petros Hadjicostas, May 06 2020 (for n >= 1)
-
PARI
{a(n)=local(X=x+x*O(x^n),A=1+X);A=(exp(X*exp(X))+exp(X/exp(X)))/2;n!*polcoeff(A,n)} -
PARI
{a(n)=n!*polcoeff(sum(m=0,n,x^m*cosh(m*x+x*O(x^n))/m!),n)} -
PARI
a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023
Formula
E.g.f.: Sum_{n>=0} x^n*cosh(n*x)/n!.
a(n) = Sum_{m=0..floor((n-1)/2)} binomial(n,2*m)*(n-2*m)^(2*m) for n >= 1. - Vladimir Kruchinin, Mar 10 2013 [Edited by Petros Hadjicostas, May 06 2020]