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G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n / Product_{k=1..n} (1 + k*x)^3.
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%I #5 Mar 12 2022 13:03:35

%S 1,1,4,31,377,6415,142252,3919208,129681162,5025119715,223662035160,

%T 11260717242863,633424125262667,39405127536106444,2688050940578533440,

%U 199621706483099855304,16038639938585081005722,1386688821351774846453155,128409360760837836935472512

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

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

%e G.f.: 1/(1-x) = 1 + 1*x/(1+x)^3 + 4*x^2/((1+x)*(1+2*x))^3 + 31*x^3/((1+x)*(1+2*x)*(1+3*x))^3 + 377*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x))^3 +...

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

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A118804, A208829, A193333.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 15 2012