OFFSET
0,3
FORMULA
a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(3*n,k) for n>0 with a(0)=1.
EXAMPLE
O.g.f.: A(x) = 1 + x + 32*x^2 + 3536*x^3 + 877221*x^4 + 394506859*x^5 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^6*x^2/(1-2^3*x)^2*exp(-2^3*x/(1-2^3*x))/2! + 3^9*x^3/(1-3^3*x)^3*exp(-3^3*x/(1-3^3*x))/3! + 4^12*x^4/(1-4^3*x)^4*exp(-4^3*x/(1-4^3*x))/4! +...
simplifies to a power series in x with integer coefficients.
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n+1, (k^3*x)^k/(1-k^3*x)^k*exp(-k^3*x/(1-k^3*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(3*n, k)))}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 08 2013
STATUS
approved