A132683 a(n) = binomial(2^n + n, n).

1, 3, 15, 165, 4845, 435897, 131115985, 138432467745, 525783425977953, 7271150092378906305, 368539102493388126164865, 68777035446753808820521420545, 47450879627176629761462147774626305

Offset: 0

Author

Paul D. Hanna, Aug 26 2007

Keywords

• nonn

Examples

			From _Paul D. Hanna_, Feb 25 2009: (Start)
G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...
A(x) = 1/(1-x) - log(1-2x)/(1-2x) + log(1-4x)^2/((1-4x)*2!) - log(1-8x)^3/((1-8x)*3!) +- ... (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50

Crossrefs

Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), this sequence (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).

Cf. A136555.

Cf. A066384. - Paul D. Hanna, Feb 25 2009

Programs

Magma

[Binomial(2^n +n, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

Maple

A132683:= n-> binomial(2^n +n,n); seq(A132683(n), n=0..20); # G. C. Greubel, Mar 14 2021

Mathematica

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

PARI

a(n)=binomial(2^n+n,n)

PARI

{a(n)=polcoeff(sum(m=0,n,(-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))*m!)),n)} \\ Paul D. Hanna, Feb 25 2009

Sage

[binomial(2^n +n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

a(n) = [x^n] 1/(1-x)^(2^n + 1).

G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)*n!). - Paul D. Hanna, Feb 25 2009

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016