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O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8*x) * x^n / n!.
5

%I #12 Nov 21 2017 19:44:31

%S 1,1,32767,47063200806,768305500780164501,75740854251732106906082250,

%T 31154086963475828638359480518580526,

%U 41929298560838945526242744414099901692285884,155440114706926165785630654089245708839702615196926765,1396002062838446082394548660243302585983358463911636390911298400

%N O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8*x) * x^n / n!.

%H G. C. Greubel, <a href="/A222528/b222528.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Stirling2(8*n, n).

%F a(n) = [x^(8*n)] (8*n)! * (exp(x) - 1)^n / n!.

%F a(n) = [x^(7*n)] 1 / Product_{k=1..n} (1-k*x).

%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^8)^k*x^k / (1 + k^8*x)^(k+1).

%F a(n) ~ n^(7*n) * 8^(8*n) / (sqrt(2*Pi*(1-c)*n) * exp(7*n) * (8-c)^(7*n) * c^n), where c = -LambertW(-8*exp(-8)). - _Vaclav Kotesovec_, May 11 2014

%e O.g.f.: A(x) = 1 + x + 32767*x^2 + 47063200806*x^3 + 768305500780164501*x^4 +...+ Stirling2(8*n, n)*x^n +...

%e where

%e A(x) = 1 + 1^8*x*exp(-1^8*x) + 2^16*exp(-2^8*x)*x^2/2! + 3^24*exp(-3^8*x)*x^3/3! + 4^32*exp(-4^8*x)*x^4/4! + 5^40*exp(-5^8*x)*x^5/5! +...

%e is a power series in x with integer coefficients.

%t Table[StirlingS2[8*n, n],{n,0,20}] (* _Vaclav Kotesovec_, May 11 2014 *)

%o (PARI) {a(n)=polcoeff(sum(k=0, n, (k^8)^k*exp(-k^8*x +x*O(x^n))*x^k/k!), n)}

%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^8)^k*x^k/(1+k^8*x +x*O(x^n))^(k+1)), n)}

%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(7*n))), 7*n)}

%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

%o {a(n) = Stirling2(8*n, n)}

%o for(n=0, 12, print1(a(n), ", "))

%Y Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222529, A222530, A217900.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 23 2013