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%I #29 Sep 08 2018 18:53:21
%S 1,1,511,2375101,45232115901,2436684974110751,299310102746948685757,
%T 72786959006434393367186463,31712979422428631132831124895809,
%U 22982258052528294182955639980819773510,26154716515862881292012777396577993781727011
%N O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5*x) * x^n / n!.
%H Vincenzo Librandi, <a href="/A217915/b217915.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = Stirling2(5*n, n).
%F a(n) = [x^(5*n)] (5*n)! * (exp(x) - 1)^n / n!.
%F a(n) = [x^(4*n)] 1 / Product_{k=1..n} (1-k*x).
%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^5)^k*x^k / (1 + k^5*x)^(k+1).
%F a(n) ~ n^(4*n)*5^(5*n) / (sqrt(2*Pi*n*(1-c)) * exp(4*n) * (5-c)^(4*n) * c^n), where c = -LambertW(-5/exp(5)) = 0.0348857682557... - _Vaclav Kotesovec_, May 23 2013
%e O.g.f.: A(x) = 1 + x + 511*x^2 + 2375101*x^3 + 45232115901*x^4 +...+ Stirling2(5*n, n)*x^n +...
%e where
%e A(x) = 1 + 1^5*x*exp(-1^5*x) + 2^10*exp(-2^5*x)*x^2/2! + 3^15*exp(-3^5*x)*x^3/3! + 4^20*exp(-4^5*x)*x^4/4! + 5^25*exp(-5^5*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Table[StirlingS2[5*n,n],{n,0,20}] (* _Vaclav Kotesovec_, May 23 2013 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,(k^5)^k*exp(-k^5*x +x*O(x^n))*x^k/k!),n)}
%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^5)^k*x^k/(1+k^5*x +x*O(x^n))^(k+1)), n)}
%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(4*n))), 4*n)}
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = Stirling2(5*n, n)}
%o for(n=0,12,print1(a(n),", "))
%o (Maxima) makelist(stirling2(5*n, n), n, 0, 10); /* _Martin Ettl_, Oct 15 2012 */
%Y Cf. A007820, A217913, A217914, A217900, A008277.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 14 2012
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