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A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7*x) * x^n / n!.

%I A222527 #10 Nov 21 2017 18:58:54
%S A222527 1,1,8191,1742343625,2998587019946701,24204004899040755811870,
%T A222527 666480349285726891499539272955,
%U A222527 50789872166903636182659702516635946082,9237419992097529135737293866043969707761346313,3590622358224471993651445012122431990834934483552661750
%N A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7*x) * x^n / n!.
%H A222527 G. C. Greubel, <a href="/A222527/b222527.txt">Table of n, a(n) for n = 0..83</a>
%F A222527 a(n) = Stirling2(7*n, n).
%F A222527 a(n) = [x^(7*n)] (7*n)! * (exp(x) - 1)^n / n!.
%F A222527 a(n) = [x^(6*n)] 1 / Product_{k=1..n} (1-k*x).
%F A222527 a(n) = 1/n! * [x^n] Sum_{k>=0} (k^7)^k*x^k / (1 + k^7*x)^(k+1).
%F A222527 a(n) ~ n^(6*n) * 7^(7*n) / (sqrt(2*Pi*(1-c)*n) * exp(6*n) * (7-c)^(6*n) * c^n), where c = -LambertW(-7*exp(-7)). - _Vaclav Kotesovec_, May 11 2014
%e A222527 O.g.f.: A(x) = 1 + x + 8191*x^2 + 1742343625*x^3 + 2998587019946701*x^4 +...+ Stirling2(7*n, n)*x^n +...
%e A222527 where
%e A222527 A(x) = 1 + 1^7*x*exp(-1^7*x) + 2^14*exp(-2^7*x)*x^2/2! + 3^21*exp(-3^7*x)*x^3/3! + 4^28*exp(-4^7*x)*x^4/4! + 5^35*exp(-5^7*x)*x^5/5! +...
%e A222527 is a power series in x with integer coefficients.
%t A222527 Table[StirlingS2[7*n, n],{n,0,20}] (* _Vaclav Kotesovec_, May 11 2014 *)
%o A222527 (PARI) {a(n)=polcoeff(sum(k=0, n, (k^7)^k*exp(-k^7*x +x*O(x^n))*x^k/k!), n)}
%o A222527 (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^7)^k*x^k/(1+k^7*x +x*O(x^n))^(k+1)), n)}
%o A222527 (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(6*n))), 6*n)}
%o A222527 (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A222527 {a(n) = Stirling2(7*n, n)}
%o A222527 for(n=0, 12, print1(a(n), ", "))
%Y A222527 Cf. A007820, A217913, A217914, A217915, A222526, A222528, A222529, A222530, A217900.
%K A222527 nonn
%O A222527 0,3
%A A222527 _Paul D. Hanna_, Feb 23 2013