A136637 - OEIS

%I #11 Jul 02 2016 07:53:26

%S 1,5,72,6089,3326498,12405917044,336474648380394,69883583587428350874,

%T 115099747754889610404191160,1536533057081060754026861201898620,

%U 168527150638482484315370462123098294514192

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

%C Equals row sums of triangle A136635.

%F G.f.: A(x) = Sum_{n>=0} log(1 + (2^n + 3^n)*x )^n / n!.

%F a(n) ~ 3^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016

%e More generally,

%e if Sum_{n>=0} log(1 + (p^n + r*q^n)*x )^n / n! = Sum_{n>=0} b(n)*x^n,

%e then b(n) = Sum_{k=0..n} C(n,k)*r^(n-k) * C(p^k*q^(n-k), n)

%e (a result due to _Vladeta Jovovic_, Jan 13 2008).

%t Table[Sum[Binomial[n,k]*Binomial[2^k*3^(n-k),n], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k*3^(n-k),n))}

%o (PARI) /* Using g.f.: */ {a(n)=polcoeff(sum(i=0,n,log(1+(2^i+3^i)*x)^i/i!),n,x)}

%Y Cf. A136635 (triangle), A014070 (main diagonal), A136393 (column 0), A136636 (column 1), A136638 (antidiagonal sums).

%K nonn

%O 0,2

%A _Vladeta Jovovic_ and _Paul D. Hanna_, Jan 15 2008