OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
a(n) = Stirling2(5*n, n).
a(n) = [x^(5*n)] (5*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(4*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^5)^k*x^k / (1 + k^5*x)^(k+1).
a(n) ~ n^(4*n)*5^(5*n) / (sqrt(2*Pi*n*(1-c)) * exp(4*n) * (5-c)^(4*n) * c^n), where c = -LambertW(-5/exp(5)) = 0.0348857682557... - Vaclav Kotesovec, May 23 2013
EXAMPLE
O.g.f.: A(x) = 1 + x + 511*x^2 + 2375101*x^3 + 45232115901*x^4 +...+ Stirling2(5*n, n)*x^n +...
where
A(x) = 1 + 1^5*x*exp(-1^5*x) + 2^10*exp(-2^5*x)*x^2/2! + 3^15*exp(-3^5*x)*x^3/3! + 4^20*exp(-4^5*x)*x^4/4! + 5^25*exp(-5^5*x)*x^5/5! +...
is a power series in x with integer coefficients.
MATHEMATICA
Table[StirlingS2[5*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 23 2013 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, (k^5)^k*exp(-k^5*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^5)^k*x^k/(1+k^5*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(4*n))), 4*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(5*n, n)}
for(n=0, 12, print1(a(n), ", "))
(Maxima) makelist(stirling2(5*n, n), n, 0, 10); /* Martin Ettl, Oct 15 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2012
STATUS
approved