A375695 Expansion of e.g.f. 1 / sqrt(1 - 2 * x * (exp(x) - 1)).

1, 0, 2, 3, 40, 185, 2436, 20797, 307616, 3869217, 66259900, 1091351261, 21671302368, 437191547377, 9981020325836, 236821065758565, 6144729994822336, 167019469703969345, 4868403452056231164, 148845363155530699789, 4822574537456548631360

Offset: 0

Author

Seiichi Manyama, Aug 24 2024

Keywords

• nonn

Crossrefs

Cf. A305404, A367880, A375687.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/Sqrt[1-2x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-2*x*(exp(x)-1))))

PARI

a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = n!*sum(k=0, n, a001147(k)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} A001147(k) * Stirling2(n-k,k)/(n-k)!.