A132683 - OEIS

%I #15 Mar 14 2021 18:44:19

%S 1,3,15,165,4845,435897,131115985,138432467745,525783425977953,

%T 7271150092378906305,368539102493388126164865,

%U 68777035446753808820521420545,47450879627176629761462147774626305

%N a(n) = binomial(2^n + n, n).

%H G. C. Greubel, <a href="/A132683/b132683.txt">Table of n, a(n) for n = 0..50</a>

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

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

%F a(n) ~ 2^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016

%e From _Paul D. Hanna_, Feb 25 2009: (Start)

%e G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...

%e 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)

%p A132683:= n-> binomial(2^n +n,n); seq(A132683(n), n=0..20); # _G. C. Greubel_, Mar 14 2021

%t Table[Binomial[2^n+n, n], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)

%o (PARI) a(n)=binomial(2^n+n,n)

%o (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

%o (Sage) [binomial(2^n +n, n) for n in (0..20)] # _G. C. Greubel_, Mar 14 2021

%o (Magma) [Binomial(2^n +n, n): n in [0..20]]; // _G. C. Greubel_, Mar 14 2021

%Y 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).

%Y Cf. A136555.

%Y Cf. A066384. - _Paul D. Hanna_, Feb 25 2009

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 26 2007