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Expansion of e.g.f. exp(sinh(x)*x) (even powers only).
5

%I #29 Apr 11 2022 16:38:32

%S 1,2,16,246,5944,202330,9099564,517447126,36048776656,3003924569778,

%T 293835907664980,33232296062419630,4291773869167401720,

%U 626311538509296801226,102365694283336181089084,18595053487766135171539590,3729223211361742071603266464

%N Expansion of e.g.f. exp(sinh(x)*x) (even powers only).

%F a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * k * a(n-k). - _Ilya Gutkovskiy_, Mar 10 2022

%t With[{nn=30},Take[CoefficientList[Series[Exp[Sinh[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Jul 31 2020 *)

%o (PARI) my(x='x+O('x^40), v=Vec(serlaplace(exp(sinh(x)*x)))); vector(#v\2, k, v[2*k-1]) \\ _Michel Marcus_, Mar 10 2022

%Y Cf. A009214, A352254.

%K nonn

%O 0,2

%A _R. H. Hardin_

%E Extended and signs tested by _Olivier GĂ©rard_, Mar 15 1997

%E Previous Mathematica program replaced by _Harvey P. Dale_, Jul 31 2020