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

0,2

Links

Table of n, a(n) for n=0..18.

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

Conjecture: a(n) = b(n-1) + 2 * Sum_{j=0..n-1} binomial(n, j) * a(j) for n > 0 where b(n) = b(n-1) + Sum_{j=0..n-1} binomial(n-1,j) * a(j+1) for n > 0 with b(0) = 1. Also it seems that b(n) = A305404(n+1). - Mikhail Kurkov, Jun 14 2026

Mathematica

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

Prog

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

Crossrefs

Cf. A005649, A375949.

Cf. A004123, A367470, A367471, A375954.

Cf. A001147, A305404.

Sequence in context: A371538 A207569 A005412 * A352647 A145350 A107888

Adjacent sequences: A375945 A375946 A375947 * A375949 A375950 A375951

Keyword

nonn,changed

Author

Seiichi Manyama, Sep 03 2024

Status

approved