cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A136637 a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).

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%I A136637 #11 Jul 02 2016 07:53:26
%S A136637 1,5,72,6089,3326498,12405917044,336474648380394,69883583587428350874,
%T A136637 115099747754889610404191160,1536533057081060754026861201898620,
%U A136637 168527150638482484315370462123098294514192
%N A136637 a(n) = Sum_{k=0..n} C(n, k) * C(2^k*3^(n-k), n).
%C A136637 Equals row sums of triangle A136635.
%F A136637 G.f.: A(x) = Sum_{n>=0} log(1 + (2^n + 3^n)*x )^n / n!.
%F A136637 a(n) ~ 3^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016
%e A136637 More generally,
%e A136637 if Sum_{n>=0} log(1 + (p^n + r*q^n)*x )^n / n! = Sum_{n>=0} b(n)*x^n,
%e A136637 then b(n) = Sum_{k=0..n} C(n,k)*r^(n-k) * C(p^k*q^(n-k), n)
%e A136637 (a result due to _Vladeta Jovovic_, Jan 13 2008).
%t A136637 Table[Sum[Binomial[n,k]*Binomial[2^k*3^(n-k),n], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)
%o A136637 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k*3^(n-k),n))}
%o A136637 (PARI) /* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+(2^i+3^i)*x)^i/i!),n,x)}
%Y A136637 Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136638 (antidiagonal sums).
%K A136637 nonn
%O A136637 0,2
%A A136637 _Vladeta Jovovic_ and _Paul D. Hanna_, Jan 15 2008