|
%I #10 Nov 21 2017 13:53:10
%S 1,1,2047,64439010,11681056634501,7713000216608565075,
%T 14204422416132896951197888,61232072982330045410678351728440,
%U 545827051514425992551826008968173372261,9173647538352903119028122246836507680995590680
%N O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6*x) * x^n / n!.
%H G. C. Greubel, <a href="/A222526/b222526.txt">Table of n, a(n) for n = 0..96</a>
%F a(n) = Stirling2(6*n, n).
%F a(n) = [x^(6*n)] (6*n)! * (exp(x) - 1)^n / n!.
%F a(n) = [x^(5*n)] 1 / Product_{k=1..n} (1-k*x).
%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^6)^k*x^k / (1 + k^6*x)^(k+1).
%F a(n) ~ n^(5*n) * 6^(6*n) / (sqrt(2*Pi*(1-c)*n) * exp(5*n) * (6-c)^(5*n) * c^n), where c = -LambertW(-6*exp(-6)). - _Vaclav Kotesovec_, May 11 2014
%e O.g.f.: A(x) = 1 + x + 2047*x^2 + 64439010*x^3 + 11681056634501*x^4 +...+ Stirling2(6*n, n)*x^n +...
%e where
%e A(x) = 1 + 1^6*x*exp(-1^6*x) + 2^12*exp(-2^6*x)*x^2/2! + 3^18*exp(-3^6*x)*x^3/3! + 4^24*exp(-4^6*x)*x^4/4! + 5^30*exp(-5^6*x)*x^5/5! +...
%e is a power series in x with integer coefficients.
%t Table[StirlingS2[6*n, n],{n,0,20}] (* _Vaclav Kotesovec_, May 11 2014 *)
%o (PARI) {a(n)=polcoeff(sum(k=0, n, (k^6)^k*exp(-k^6*x +x*O(x^n))*x^k/k!), n)}
%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^6)^k*x^k/(1+k^6*x +x*O(x^n))^(k+1)), n)}
%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(5*n))), 5*n)}
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = Stirling2(6*n, n)}
%o for(n=0, 12, print1(a(n), ", "))
%Y Cf. A007820, A217913, A217914, A217915, A222527, A222528, A222529, A222530, A217900.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 23 2013
| 