OFFSET
0,3
COMMENTS
Generally, for p>=2 is StirlingS2(p*n,n) asymptotic to n^((p-1)*n) * p^(p*n) / (sqrt(2*Pi*(1-c)*n) * exp((p-1)*n) * (p-c)^((p-1)*n) * c^n), where c = -LambertW(-p*exp(-p)). - Vaclav Kotesovec, May 11 2014
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..60
FORMULA
a(n) = Stirling2(10*n, n).
a(n) = [x^(10*n)] (10*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(9*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^10)^k*x^k / (1 + k^10*x)^(k+1).
a(n) ~ n^(9*n) * 10^(10*n) / (sqrt(2*Pi*(1-c)*n) * exp(9*n) * (10-c)^(9*n) * c^n), where c = -LambertW(-10*exp(-10)). - Vaclav Kotesovec, May 11 2014
EXAMPLE
O.g.f.: A(x) = 1 + x + 524287*x^2 + 34314651811530*x^3 + 50369882873307917364901*x^4 +...+ Stirling2(10*n, n)*x^n +...
where
A(x) = 1 + 1^10*x*exp(-1^10*x) + 2^20*exp(-2^10*x)*x^2/2! + 3^30*exp(-3^10*x)*x^3/3! + 4^40*exp(-4^10*x)*x^4/4! + 5^50*exp(-5^10*x)*x^5/5! +...
is a power series in x with integer coefficients.
MATHEMATICA
Table[StirlingS2[10*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, (k^10)^k*exp(-k^10*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^1)^k*x^k/(1+k^10*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(9*n))), 9*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(10*n, n)}
for(n=0, 12, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2013
STATUS
approved