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

%I A009229 #43 Feb 26 2022 10:51:58
%S A009229 1,1,1,5,17,57,337,1661,9697,68529,451809,3570613,29324913,247557481,
%T A009229 2348275633,22399292077,229227939521,2476740370529,27378158737089,
%U A009229 323000223133285,3912000089144273,49350756905258905,649905519911431953,8776449485870211549
%N A009229 Expansion of e.g.f. exp(sinh(x)*cosh(x)).
%H A009229 Alois P. Heinz, <a href="/A009229/b009229.txt">Table of n, a(n) for n = 0..528</a> (first 126 terms from Vincenzo Librandi)
%H A009229 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A009229 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
%F A009229 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A009229 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.
%F A009229 (a(n)/n!)^(1/n) ~ 2*(exp(1/LambertW(4*n))/LambertW(4*n)).
%F A009229 (End)
%F A009229 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
%p A009229 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
%p A009229       a(n-j)*binomial(n-1, j-1)*2^(j-1), 0), j=1..n))
%p A009229     end:
%p A009229 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 22 2021
%t A009229 With[{nn=30},CoefficientList[Series[Exp[Sinh[x]Cosh[x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 23 2011 *)
%o A009229 (Maxima)
%o A009229 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 */
%o A009229 (PARI) x='x+O('x^66); /* that many terms */
%o A009229 Vec(serlaplace(exp(sinh(x)*cosh(x)))) /* show terms */ /* _Joerg Arndt_, May 29 2011 */
%K A009229 nonn,easy
%O A009229 0,4
%A A009229 _R. H. Hardin_
%E A009229 Extended and signs tested by _Olivier Gérard_, Mar 15 1997
%E A009229 Corrected definition, _Joerg Arndt_, May 29 2011