A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

1, 3, 18, 153, 1683, 22698, 362403, 6683463, 139787568, 3269240883, 84535585263, 2394699999948, 73749495626253, 2453332830142743, 87667856626175298, 3349116499958627733, 136209377351085310863, 5875794769594996985778, 267968680043585007829383

Offset: 0

Author

Seiichi Manyama, Sep 03 2024

Keywords

• nonn

Crossrefs

Cf. A005649, A375949.

Cf. A004123, A367470, A367471, A375954.

Cf. A001147.

Programs

Mathematica

nmax=18; CoefficientList[Series[1 / (3 - 2 * Exp[x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

PARI

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

Formula

a(n) = Sum_{k=0..n} A001147(k+1) * Stirling2(n,k).

a(n) ~ 2^(3/2) * n^(n+1) / (3^(3/2) * log(3/2)^(n + 3/2) * exp(n)). - Vaclav Kotesovec, May 20 2025