A009236 E.g.f. exp(sinh(x)^2) (even powers only).

1, 2, 20, 392, 12560, 579872, 36034880, 2874676352, 284538241280, 34058188677632, 4831480473359360, 799233222752602112, 152126941229960990720, 32947584100184816033792, 8042650107769696199720960, 2194728130327915760239542272, 664779526701915421094019399680

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..264

Formula

a(n) = Sum_{k=1..n}(Sum_{i=0..2*k} (-1)^i*(k-i)^(2*n)*binomial(2*k,i))*(2^(2*(n-k)) /k!)), n>0, a(0)=1. - Vladimir Kruchinin, Jun 06 2011

a(n) = 4^n*Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*B(k, 1/4)*B(2*n-k, 1/4) where B(n, x) are the Bell polynomials. - Peter Luschny, Sep 10 2017

Maple

A009236 := n -> 4^n*(add(binomial(2*n, k)*(-1)^k*BellB(k, 1/4)*BellB(2*n-k, 1/4), k = 0..2*n)): seq(A009236(n), n=0..16); # Peter Luschny, Sep 10 2017
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*2^(2*j-1), j=1..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..16);  # Alois P. Heinz, Jun 21 2021

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-2*j]*Binomial[n-1, 2*j-1]*2^(2*j-1), {j, 1, n/2}]];
a[n_] := b[2*n];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, May 21 2022, after Alois P. Heinz *)

Prog

(Maxima) a(n):=sum(sum((-1)^i*(k-i)^(2*n)*binomial(2*k, i), i, 0, 2*k)*(2^(2*(n-k))/k!), k, 1, n); /* Vladimir Kruchinin, Jun 06 2011 */
(PARI) my(x='x+O('x^40), v = Vec(serlaplace(exp(sinh(x)^2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, May 21 2022

Crossrefs

Sequence in context: A376391 A376394 A218306 * A078698 A090728 A210896

Adjacent sequences: A009233 A009234 A009235 * A009237 A009238 A009239

Keyword

nonn

Author

R. H. Hardin

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

Status

approved