A134173 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2^k,n).

1, 3, 8, 68, 2106, 223776, 80532200, 98945392200, 421225839051260, 6310402120912239968, 337401124757628967733136, 65171905481441631827737564000, 45944096973025484590366745753166436

Offset: 0

Author

Paul D. Hanna and Vladeta Jovovic, Jan 13 2008

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..59

Programs

Maple

a:=proc(n) options operator, arrow: sum(binomial(n,k)*binomial(2^k,n),k=0..n) end proc: seq(a(n),n=0..13); # Emeric Deutsch, Jan 27 2008

Mathematica

Table[Sum[Binomial[n,k]*Binomial[2^k,n],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI

for(n=0,25, print1(sum(k=0,n, binomial(n,k)*binomial(2^k,n)), ", ")) \\ G. C. Greubel, Mar 21 2017

Formula

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

From Vaclav Kotesovec, Jul 02 2016: (Start)

a(n) ~ binomial(2^n,n).

a(n) ~ 2^(n^2) / n!.

a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).

(End)

Extensions

More terms from Emeric Deutsch, Jan 27 2008