A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4*x) * x^n / n!.

1, 1, 127, 86526, 171798901, 749206090500, 6090236036084530, 82892803728383735268, 1751346256720122175776157, 54294340536065700496358447625, 2364684125291482936353925428946680, 139762001313639974628848043262243505970, 10897986831117690497797320098390628446479030

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

Cf. A007820, A217913, A217915, A217900, A008277.

Sequence in context: A069434 A175633 A175825 * A203798 A351134 A195218

Adjacent sequences: A217911 A217912 A217913 * A217915 A217916 A217917

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2012

Status

approved