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A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9*x) * x^n / n!.

%I A222529 #11 Nov 21 2017 19:44:37
%S A222529 1,1,131071,1270865805301,196740254364198919901,
%T A222529 236795997997922560392792426501,
%U A222529 1454443713270449746545892977574122129433,34559048315358253352594346952765431711799794270765,2610516895723221966171633379256064857587637240616032299710417
%N A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9*x) * x^n / n!.
%H A222529 G. C. Greubel, <a href="/A222529/b222529.txt">Table of n, a(n) for n = 0..66</a>
%F A222529 a(n) = Stirling2(9*n, n).
%F A222529 a(n) = [x^(9*n)] (9*n)! * (exp(x) - 1)^n / n!.
%F A222529 a(n) = [x^(8*n)] 1 / Product_{k=1..n} (1-k*x).
%F A222529 a(n) = 1/n! * [x^n] Sum_{k>=0} (k^9)^k*x^k / (1 + k^9*x)^(k+1).
%F A222529 a(n) ~ n^(8*n) * 9^(9*n) / (sqrt(2*Pi*(1-c)*n) * exp(8*n) * (9-c)^(8*n) * c^n), where c = -LambertW(-9*exp(-9)). - _Vaclav Kotesovec_, May 11 2014
%e A222529 O.g.f.: A(x) = 1 + x + 131071*x^2 + 1270865805301*x^3 + 196740254364198919901*x^4 +...+ Stirling2(9*n, n)*x^n +...
%e A222529 where
%e A222529 A(x) = 1 + 1^9*x*exp(-1^9*x) + 2^18*exp(-2^9*x)*x^2/2! + 3^27*exp(-3^9*x)*x^3/3! + 4^36*exp(-4^9*x)*x^4/4! + 5^45*exp(-5^9*x)*x^5/5! +...
%e A222529 is a power series in x with integer coefficients.
%t A222529 Table[StirlingS2[9*n, n],{n,0,20}] (* _Vaclav Kotesovec_, May 11 2014 *)
%o A222529 (PARI) {a(n)=polcoeff(sum(k=0, n, (k^9)^k*exp(-k^9*x +x*O(x^n))*x^k/k!), n)}
%o A222529 (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^9)^k*x^k/(1+k^9*x +x*O(x^n))^(k+1)), n)}
%o A222529 (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(8*n))), 8*n)}
%o A222529 (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A222529 {a(n) = Stirling2(9*n, n)}
%o A222529 for(n=0, 12, print1(a(n), ", "))
%Y A222529 Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222528, A222530, A217900.
%K A222529 nonn
%O A222529 0,3
%A A222529 _Paul D. Hanna_, Feb 23 2013