A133991 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2^k+n-1,n).

1, 3, 17, 193, 5427, 463023, 134675759, 139917028089, 527871326293913, 7281357469833220843, 368715613115281663650597, 68787958348542935934247206953, 47453320297069210448891035137347047

Offset: 0

Author

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..59

Crossrefs

Cf. A133990, A134173, A134174.

Programs

Mathematica

Table[Sum[(-1)^(n-k) * StirlingS1[n, k]*(2^k + 1)^n, {k, 0, n}]/n!, {n, 0, 12}] (* Vaclav Kotesovec, Jun 08 2019 *)

PARI

a(n) = sum(k=0, n, binomial(n, k)*binomial(2^k+n-1, n)); \\ Seiichi Manyama, Feb 24 2023

Formula

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*(2^k+1)^n.

G.f.: Sum_{n>=0} (-log(1 - (2^n+1)*x))^n/n!.

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 08 2019

Extensions

More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 14 2008