A009229 Expansion of e.g.f. exp(sinh(x)*cosh(x)).

1, 1, 1, 5, 17, 57, 337, 1661, 9697, 68529, 451809, 3570613, 29324913, 247557481, 2348275633, 22399292077, 229227939521, 2476740370529, 27378158737089, 323000223133285, 3912000089144273, 49350756905258905, 649905519911431953, 8776449485870211549

Offset: 0

Author

R. H. Hardin

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..528 (first 126 terms from Vincenzo Librandi)
• Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
      a(n-j)*binomial(n-1, j-1)*2^(j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2021

Mathematica

With[{nn=30},CoefficientList[Series[Exp[Sinh[x]Cosh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 23 2011 *)

Maxima

a(n):=sum(((sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k))*2^(n-2*k))/k!,k,1,n); /* Vladimir Kruchinin, May 29 2011 */

PARI

x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(sinh(x)*cosh(x)))) /* show terms */ /* Joerg Arndt, May 29 2011 */

Formula

a(n) = sum(k=1..n, ((sum(i=0..k, (-1)^i*(k-2*i)^n*binomial(k,i)))*2^(n-2*k))/k!), n>0, a(0)=1. - Vladimir Kruchinin, May 29 2011

From Vaclav Kotesovec, Aug 06 2014: (Start)

a(n) ~ exp(cosh(r)*sinh(r)-n) * n^(n+1/2) / (sqrt(cosh(2*r) + 2*r*sinh(2*r)) * r^(n+1/2)), where r is the root of the equation r*(cosh(r)^2 + sinh(r)^2) = n.

(a(n)/n!)^(1/n) ~ 2*(exp(1/LambertW(4*n))/LambertW(4*n)).

(End)

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 4^k * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

Corrected definition, Joerg Arndt, May 29 2011