OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Stirling2(4*n, n).
a(n) = [x^(4*n)] (4*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(3*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^4)^k*x^k / (1 + k^4*x)^(k+1).
a(n) ~ 2^(8*n)*n^(3*n)/(sqrt(2*Pi*n*(1-c))*c^n*exp(3*n)*(4-c)^(3*n)), where c = -LambertW(-4/exp(4)) = 0.07930960512711... - Vaclav Kotesovec, May 23 2013
EXAMPLE
O.g.f.: A(x) = 1 + x + 127*x^2 + 86526*x^3 + 171798901*x^4 +...+ Stirling2(4*n,n)*x^n + ...
where
A(x) = 1 + 1^4*x*exp(-1^4*x) + 2^8*exp(-2^4*x)*x^2/2! + 3^12*exp(-3^4*x)*x^3/3! + 4^16*exp(-4^4*x)*x^4/4! + 5^20*exp(-5^4*x)*x^5/5! + ...
is a power series in x with integer coefficients.
MATHEMATICA
Table[StirlingS2[4*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, (k^4)^k*exp(-k^4*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^4)^k*x^k/(1+k^4*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(3*n))), 3*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(4*n, n)}
for(n=0, 12, print1(a(n), ", "))
(Maxima) makelist(stirling2(4*n, n), n, 0, 12); /* Martin Ettl, Oct 15 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2012
STATUS
approved