A222053 O.g.f.: Sum_{n>=0} (n^3*x)^n/(1-n^3*x)^n * exp(-n^3*x/(1-n^3*x)) / n!.

1, 1, 32, 3536, 877221, 394506859, 284110844070, 302350295364613, 449340338669205876, 894210483750815778132, 2306748823711254973903838, 7516588630649080782251419791, 30292392269310179039574629318038, 148358895760995636729844370111255773

Offset

0,3

Links

Table of n, a(n) for n=0..13.

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

Cf. A134055, A174845, A222054, A217913, A008277.

Sequence in context: A123802 A264057 A221659 * A264126 A304401 A328275

Adjacent sequences: A222050 A222051 A222052 * A222054 A222055 A222056

Keyword

nonn

Author

Paul D. Hanna, Mar 08 2013

Status

approved