A217913 - OEIS

%I #45 Sep 08 2018 18:53:01

%S 1,1,31,3025,611501,210766920,110687251039,82310957214948,

%T 82318282158320505,106563273280541795575,173373343599189364594756,

%U 346289681454731077633095526,833091176987705031151553054843,2376102520162485084539597049185710

%N O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3*x) * x^n / n!.

%H Vincenzo Librandi, <a href="/A217913/b217913.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Stirling2(3*n, n).

%F a(n) = [x^(3*n)] (3*n)! * (exp(x) - 1)^n / n!.

%F a(n) = [x^(2*n)] 1 / Product_{k=1..n} (1-k*x).

%F a(n) = 1/n! * [x^n] Sum_{k>=0} (k^3)^k*x^k / (1 + k^3*x)^(k+1).

%F a(n) ~ 9^n*exp(n*(c+1))*n^(2*n)/((c+3)^(2*n)*sqrt(2*Pi*(c+1)*n)), where c = -0.1785606278779211... = LambertW(-3/exp(3)) = A226750. - _Vaclav Kotesovec_, Feb 28 2013

%e O.g.f.: A(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + ... + Stirling2(3*n, n)*x^n + ...

%e where

%e A(x) = 1 + 1^3*x*exp(-1^3*x) + 2^6*exp(-2^3*x)*x^2/2! + 3^9*exp(-3^3*x)*x^3/3! + 4^12*exp(-4^3*x)*x^4/4! + 5^15*exp(-5^3*x)*x^5/5! + ...

%e simplifies to a power series in x with integer coefficients.

%t Table[StirlingS2[3*n,n],{n,0,20}] (* _Vaclav Kotesovec_, Feb 28 2013 *)

%o (PARI) {a(n)=polcoeff(sum(k=0,n,(k^3)^k*exp(-k^3*x +x*O(x^n))*x^k/k!),n)}

%o (PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^3)^k*x^k/(1+k^3*x +x*O(x^n))^(k+1)), n)}

%o (PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n)}

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

%o {a(n) = Stirling2(3*n, n)}

%o for(n=0,20,print1(a(n),", "))

%o (Maxima) makelist(stirling2(3*n, n), n, 0, 13); /* _Martin Ettl_, Oct 15 2012 */

%Y Cf. A219228, A007820, A217914, A217915, A217900, A008277.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 14 2012