A133990 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k + n - 1,n).

1, 1, 5, 71, 2747, 306861, 106709627, 123122238887, 492425723170553, 7012142056418141897, 361269845371107759765065, 68033187103968192731087467135, 47171609221094330538117045468744655

Offset: 0

Author

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Keywords

• easy
• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 71*x^3 + 2747*x^4 + 306861*x^5 +...
where
A(x) = 1 - log(1-x) + log(1-3*x)^2/2! - log(1-7*x)^3/3! + log(1-15*x)^4/4! - log(1-31*x)^5/5! + log(1-63*x)^6/6! - log(1-127*x)^7/7! + log(1-255*x)^8/8! +...
		

Links

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

Crossrefs

Cf. A134174.

Programs

Maple

A133990 := proc(n) add((-1)^(n-k)*binomial(n,k)*binomial(2^k+n-1,n),k=0..n) ; end: seq(A133990(n),n=0..15) ; # R. J. Mathar, Jan 30 2008

Mathematica

Table[Sum[(-1)^(n-k) Binomial[n,k]Binomial[2^k+n-1,n],{k,0,n}], {n,0,15}] (* Harvey P. Dale, Nov 24 2011 *)

PARI

{a(n)=local(A=1,X=x+x*O(x^n)); A=sum(k=0,n,log(1/(1-(2^k-1)*X))^k/k!); polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=1/n!*sum(k=0,n,(-1)^(n-k)*Stirling1(n,k)*(2^k-1)^n)}
for(n=0, 20, print1(a(n), ", "))

Formula

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

a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * (2^k-1)^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 R. J. Mathar, Jan 30 2008