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%I A132684 #17 Mar 14 2021 18:44:25
%S A132684 1,4,21,220,5985,501942,143218999,145944307080,542150225230185,
%T A132684 7398714129087308170,372134605932348010322571,
%U A132684 69146263065062394421802892300,47589861944854471977019273909187085
%N A132684 a(n) = binomial(2^n + n + 1, n).
%H A132684 G. C. Greubel, <a href="/A132684/b132684.txt">Table of n, a(n) for n = 0..50</a>
%F A132684 a(n) = [x^n] 1/(1-x)^(2^n + 2).
%F A132684 G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)^2*n!). - _Paul D. Hanna_, Feb 25 2009
%F A132684 a(n) ~ 2^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016
%e A132684 From _Paul D. Hanna_, Feb 25 2009: (Start)
%e A132684 G.f.: A(x) = 1 + 4*x + 21*x^2 + 220*x^3 + 5985*x^4 + 501942*x^5 +...
%e A132684 A(x) = 1/(1-x)^2 - log(1-2x)/(1-2x)^2 + log(1-4x)^2/((1-4x)^2*2!) - log(1-8x)^3/((1-8x)^2*3!) +- ... (End)
%p A132684 A132684:= n-> binomial(2^n +n+1, n); seq(A132684(n), n=0..20); # _G. C. Greubel_, Mar 14 2021
%t A132684 Table[Binomial[2^n+n+1,n],{n,0,20}] (* _Harvey P. Dale_, Nov 10 2011 *)
%o A132684 (PARI) a(n)=binomial(2^n+n+1,n)
%o A132684 (PARI) {a(n)=polcoeff(sum(m=0,n,(-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))^2*m!)),n)} \\ _Paul D. Hanna_, Feb 25 2009
%o A132684 (Sage) [binomial(2^n +n+1, n) for n in (0..20)] # _G. C. Greubel_, Mar 14 2021
%o A132684 (Magma) [Binomial(2^n +n+1, n): n in [0..20]]; // _G. C. Greubel_, Mar 14 2021
%Y A132684 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), A132683 (1,0), this sequence (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
%Y A132684 Cf. A136555.
%Y A132684 Cf. A066384. - _Paul D. Hanna_, Feb 25 2009
%K A132684 nonn
%O A132684 0,2
%A A132684 _Paul D. Hanna_, Aug 26 2007

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