A352254 -id:A352254 - cp's OEIS frontend

A009233 Expansion of e.g.f. exp(sinh(x)*x) (even powers only).

1, 2, 16, 246, 5944, 202330, 9099564, 517447126, 36048776656, 3003924569778, 293835907664980, 33232296062419630, 4291773869167401720, 626311538509296801226, 102365694283336181089084, 18595053487766135171539590, 3729223211361742071603266464

Offset: 0

R. H. Hardin

nonn

Cf. A009214, A352254.

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 *)

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

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

Extended and signs tested by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Jul 31 2020

A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

Original entry on oeis.org

1, 1, 8, 153, 5492, 316625, 26774622, 3121729709, 479962730648, 94087054172673, 22904161764512570, 6778870099212235805, 2397161662661680925364, 998186321121004312238513, 483430830256916593106991782, 269435322393253822641626419725, 171224984800186115316322226731952

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A006153, A006154, A094088, A205571, A352251, A352254.

nmax = 32; Take[CoefficientList[Series[1/(1 - x Sinh[x]/2), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

cp's OEIS Frontend

A009233 Expansion of e.g.f. exp(sinh(x)*x) (even powers only).

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