A198952 - OEIS

%I #16 May 06 2012 23:19:33

%S 1,1,3,45,3267,991845,1155605211,4910640919821,73614877173054099,

%T 3802910817051064124469,665332303024345700007225099,

%U 388955052253927480089824057425437,751710022839628223241451188902204177091

%N G.f.: Sum_{n>=0} n! * 3^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + k*3^k*x).

%C Compare the g.f. to the identities:

%C (1) 1/(1-x) = Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + k*x).

%C (2) 1+x = Sum_{n>=0} 3^(n*(n-1)/2) * x^n / Product_{k=1..n} (1 + 3^k*x).

%e G.f.: A(x) = 1 + x + 3*x^2 + 45*x^3 + 3267*x^4 + 991845*x^5 + 1155605211*x^6 +...

%e such that

%e A(x) = 1 + x/(1+3*x) + 2!*3^1*x^2/((1+1*3*x)*(1+2*9*x)) + 3!*3^3*x^3/((1+1*3*x)*(1+2*9*x)*(1+3*27*x)) + 4!*3^6*x^4/((1+1*3*x)*(1+2*9*x)*(1+3*27*x)*(1+4*81*x)) +...

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

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A182507.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 06 2012