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

0,3

Links

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

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)

Example

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! +...

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A134174.

Sequence in context: A347989 A193436 A193501 * A326881 A120808 A092204

Adjacent sequences: A133987 A133988 A133989 * A133991 A133992 A133993

Keyword

easy,nonn

Author

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Extensions

More terms from R. J. Mathar, Jan 30 2008

Status

approved