A196559 - OEIS

%I #10 Aug 25 2024 14:01:09

%S 1,1,3,12,65,384,2197,14078,94739,670612,4899280,36645899,281037158,

%T 2197679518,17489660228,141241307806,1155345218645,9559672712389,

%U 79905432682918,674005489358155,5731854529045978,49105864505432392,423531623342726441

%N G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k]^2 * x^n/n ).

%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 65*x^4 + 384*x^5 + 2197*x^6 +...

%e where

%e log(A(x)) = (1 + x)^2*x + (1+2^3*x+x^2)^2*x^2/2 + (1+3^3*x+3^3*x^2+x^3)^2*x^3/3 + (1+4^3*x+6^3*x^2+4^3*x^3+x^4)^2*x^4/4 +...

%e More explicitly,

%e log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 205*x^4/4 + 1506*x^5/5 + 10016*x^6/6 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^3*x^k)^2*x^m/m)+x*O(x^n)), n)}

%Y Cf. A166896, A180718, A196560.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 03 2011