A352254 -id:A352254

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A009233

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..16.

FORMULA

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A009214, A352254.

KEYWORD

nonn

AUTHOR

R. H. Hardin

EXTENSIONS

Extended and signs tested by Olivier Gérard, Mar 15 1997

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

STATUS

approved

A352253

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..16.

FORMULA

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

MATHEMATICA

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}]

PROG

(PARI) 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