cp's OEIS Frontend

A222053 O.g.f.: Sum_{n>=0} (n^3*x)^n/(1-n^3*x)^n * exp(-n^3*x/(1-n^3*x)) / n!.

%I A222053 #7 Jun 06 2025 19:07:04
%S A222053 1,1,32,3536,877221,394506859,284110844070,302350295364613,
%T A222053 449340338669205876,894210483750815778132,2306748823711254973903838,
%U A222053 7516588630649080782251419791,30292392269310179039574629318038,148358895760995636729844370111255773
%N A222053 O.g.f.: Sum_{n>=0} (n^3*x)^n/(1-n^3*x)^n * exp(-n^3*x/(1-n^3*x)) / n!.
%F A222053 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(3*n,k) for n>0 with a(0)=1.
%e A222053 O.g.f.: A(x) = 1 + x + 32*x^2 + 3536*x^3 + 877221*x^4 + 394506859*x^5 +...
%e A222053 where
%e A222053 A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^6*x^2/(1-2^3*x)^2*exp(-2^3*x/(1-2^3*x))/2! + 3^9*x^3/(1-3^3*x)^3*exp(-3^3*x/(1-3^3*x))/3! + 4^12*x^4/(1-4^3*x)^4*exp(-4^3*x/(1-4^3*x))/4! +...
%e A222053 simplifies to a power series in x with integer coefficients.
%o A222053 (PARI) {a(n)=polcoeff(sum(k=0, n+1, (k^3*x)^k/(1-k^3*x)^k*exp(-k^3*x/(1-k^3*x+x*O(x^n)))/k!), n)}
%o A222053 for(n=0, 25, print1(a(n), ", "))
%o A222053 (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A222053 {a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(3*n, k)))}
%o A222053 for(n=0, 25, print1(a(n), ", "))
%Y A222053 Cf. A134055, A174845, A222054, A217913, A008277.
%K A222053 nonn
%O A222053 0,3
%A A222053 _Paul D. Hanna_, Mar 08 2013