A009599 Expansion of e.g.f. sinh(sinh(x)*exp(x)).

0, 1, 2, 5, 20, 117, 782, 5441, 39496, 306921, 2602682, 24116413, 241121564, 2561633245, 28613237382, 334511450617, 4089814554384, 52302564139985, 699179303859698, 9751200460426357, 141494250613386916

Offset: 0

Author

R. H. Hardin

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A065143.

Programs

Magma

[(&+[2^(n-k)*StirlingSecond(n,k)*(1 - (-1)^k)/2: k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 22 2018

Mathematica

Table[Sum[StirlingS2[n, k]*(1-(-1)^k)/2*2^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 06 2014 after Vladeta Jovovic *)
Table[(BellB[n, 1/2] - BellB[n, -1/2]) 2^(n-1), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
With[{nn=20},CoefficientList[Series[Sinh[Sinh[x]Exp[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jun 02 2017 *)

PARI

x='x+O('x^30); concat([0], Vec(serlaplace(sinh(sinh(x)*exp(x))))) \\ G. C. Greubel, Jan 22 2018

Formula

a(n) = Sum_{k=0..n} Stirling2(n, k)*(1-(-1)^k)/2*2^(n-k). - Vladeta Jovovic, Sep 26 2003

G.f.: Sum_{k>=0} x^(2*k+1)/Product_{i=0..2*k+1} (1 - 2*i*x). - Sergei N. Gladkovskii, Jan 06 2013

G.f.: x/( G(0)-x^2 ) where G(k) = x^2 + (4*x*k-1)*(4*x*k+2*x-1) - x^2*(4*x*k-1)*(4*x*k+2*x-1)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 06 2013

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997