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A222054 O.g.f.: Sum_{n>=0} (n^4*x)^n/(1-n^4*x)^n * exp(-n^4*x/(1-n^4*x)) / n!.

%I A222054 #6 Jun 06 2025 19:17:15
%S A222054 1,1,128,90621,193322261,933620289929,8632521193856886,
%T A222054 136885314823146617517,3443427946573913689696192,
%U A222054 129667338445150206244162849650,6988095504452769015520539806767120,520011535068804196524689647521015780176
%N A222054 O.g.f.: Sum_{n>=0} (n^4*x)^n/(1-n^4*x)^n * exp(-n^4*x/(1-n^4*x)) / n!.
%F A222054 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(4*n,k) for n>0 with a(0)=1.
%e A222054 O.g.f.: A(x) = 1 + x + 128*x^2 + 90621*x^3 + 193322261*x^4 +...
%e A222054 where
%e A222054 A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^8*x^2/(1-2^4*x)^2*exp(-2^4*x/(1-2^4*x))/2! + 3^12*x^3/(1-3^4*x)^3*exp(-3^4*x/(1-3^4*x))/3! + 4^16*x^4/(1-4^4*x)^4*exp(-4^4*x/(1-4^4*x))/4! +...
%e A222054 simplifies to a power series in x with integer coefficients.
%o A222054 (PARI) {a(n)=polcoeff(sum(k=0, n+1, (k^4*x)^k/(1-k^4*x)^k*exp(-k^4*x/(1-k^4*x+x*O(x^n)))/k!), n)}
%o A222054 for(n=0, 20, print1(a(n), ", "))
%o A222054 (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A222054 {a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(4*n, k)))}
%o A222054 for(n=0, 20, print1(a(n), ", "))
%Y A222054 Cf. A134055, A174845, A222053, A217913, A008277.
%K A222054 nonn
%O A222054 0,3
%A A222054 _Paul D. Hanna_, Mar 08 2013