A217913 O.g.f.: Sum_{n>=0} (n^3)^n * exp(-n^3*x) * x^n / n!.
1, 1, 31, 3025, 611501, 210766920, 110687251039, 82310957214948, 82318282158320505, 106563273280541795575, 173373343599189364594756, 346289681454731077633095526, 833091176987705031151553054843, 2376102520162485084539597049185710
Offset: 0
Keywords
Examples
O.g.f.: A(x) = 1 + x + 31*x^2 + 3025*x^3 + 611501*x^4 + ... + Stirling2(3*n, n)*x^n + ... where 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! + ... simplifies to a power series in x with integer coefficients.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
Programs
-
Mathematica
Table[StirlingS2[3*n,n],{n,0,20}] (* Vaclav Kotesovec, Feb 28 2013 *) -
Maxima
makelist(stirling2(3*n, n), n, 0, 13); /* Martin Ettl, Oct 15 2012 */
-
PARI
{a(n)=polcoeff(sum(k=0,n,(k^3)^k*exp(-k^3*x +x*O(x^n))*x^k/k!),n)} -
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)} -
PARI
{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n)} -
PARI
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)} {a(n) = Stirling2(3*n, n)} for(n=0,20,print1(a(n),", "))
Formula
a(n) = Stirling2(3*n, n).
a(n) = [x^(3*n)] (3*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(2*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^3)^k*x^k / (1 + k^3*x)^(k+1).
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