|
%I #16 May 11 2014 16:41:33
%S 1,1,31,86526,45232115901,7713000216608565075,
%T 666480349285726891499539272955,
%U 41929298560838945526242744414099901692285884,2610516895723221966171633379256064857587637240616032299710417
%N a(n) = Stirling2(n^2+n, n).
%H Paul D. Hanna, <a href="/A218142/b218142.txt">Table of n, a(n) for n = 0..25</a>
%F a(n) = [x^n] Sum_{k>=0} k^((n+1)*k) * exp(-k^(n+1)*x) * x^k / k!.
%F a(n) = [x^(n^2)] 1 / Product_{k=1..n} (1-k*x).
%F a(n) ~ n^(n^2+n)/n!. - _Vaclav Kotesovec_, May 11 2014
%e O.g.f.: A(x) = 1 + x + 31*x^2 + 86526*x^3 + 45232115901*x^4 +...
%t Table[StirlingS2[n^2+n, n],{n,0,10}] (* _Vaclav Kotesovec_, May 11 2014 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,(k^(n+1))^k*exp(-k^(n+1)*x +x*O(x^n))*x^k/k!),n)}
%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2))), n^2)}
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = Stirling2(n^2+n, n)}
%o for(n=0, 10, print1(a(n), ", "))
%o (Maxima) makelist(stirling2(n^2+n,n),n,0,30 ); /* _Martin Ettl_, Oct 21 2012 */
%Y Cf. A008277, A218141, A218143, A007820, A217913, A217914, A217915.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 21 2012
| 