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%I #34 Sep 08 2018 18:53:13
%S 1,1,127,86526,171798901,749206090500,6090236036084530,
%T 82892803728383735268,1751346256720122175776157,
%U 54294340536065700496358447625,2364684125291482936353925428946680,139762001313639974628848043262243505970,10897986831117690497797320098390628446479030
%N O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4*x) * x^n / n!.
%H Vincenzo Librandi, <a href="/A217914/b217914.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = Stirling2(4*n, n).
%F a(n) = [x^(4*n)] (4*n)! * (exp(x) - 1)^n / n!.
%F a(n) = [x^(3*n)] 1 / Product_{k=1..n} (1-k*x).
%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^4)^k*x^k / (1 + k^4*x)^(k+1).
%F a(n) ~ 2^(8*n)*n^(3*n)/(sqrt(2*Pi*n*(1-c))*c^n*exp(3*n)*(4-c)^(3*n)), where c = -LambertW(-4/exp(4)) = 0.07930960512711... - _Vaclav Kotesovec_, May 23 2013
%e O.g.f.: A(x) = 1 + x + 127*x^2 + 86526*x^3 + 171798901*x^4 +...+ Stirling2(4*n,n)*x^n + ...
%e where
%e A(x) = 1 + 1^4*x*exp(-1^4*x) + 2^8*exp(-2^4*x)*x^2/2! + 3^12*exp(-3^4*x)*x^3/3! + 4^16*exp(-4^4*x)*x^4/4! + 5^20*exp(-5^4*x)*x^5/5! + ...
%e is a power series in x with integer coefficients.
%t Table[StirlingS2[4*n,n],{n,0,20}] (* _Vaclav Kotesovec_, May 23 2013 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,(k^4)^k*exp(-k^4*x +x*O(x^n))*x^k/k!),n)}
%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^4)^k*x^k/(1+k^4*x +x*O(x^n))^(k+1)), n)}
%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(3*n))), 3*n)}
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = Stirling2(4*n, n)}
%o for(n=0,12,print1(a(n),", "))
%o (Maxima) makelist(stirling2(4*n, n), n, 0, 12); /* _Martin Ettl_, Oct 15 2012 */
%Y Cf. A007820, A217913, A217915, A217900, A008277.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2012
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