A222529 O.g.f.: Sum_{n>=0} (n^9)^n * exp(-n^9*x) * x^n / n!.

1, 1, 131071, 1270865805301, 196740254364198919901, 236795997997922560392792426501, 1454443713270449746545892977574122129433, 34559048315358253352594346952765431711799794270765, 2610516895723221966171633379256064857587637240616032299710417

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..66

Formula

a(n) = Stirling2(9*n, n).

a(n) = [x^(9*n)] (9*n)! * (exp(x) - 1)^n / n!.

a(n) = [x^(8*n)] 1 / Product_{k=1..n} (1-k*x).

a(n) = 1/n! * [x^n] Sum_{k>=0} (k^9)^k*x^k / (1 + k^9*x)^(k+1).

a(n) ~ n^(8*n) * 9^(9*n) / (sqrt(2*Pi*(1-c)*n) * exp(8*n) * (9-c)^(8*n) * c^n), where c = -LambertW(-9*exp(-9)). - Vaclav Kotesovec, May 11 2014

Example

O.g.f.: A(x) = 1 + x + 131071*x^2 + 1270865805301*x^3 + 196740254364198919901*x^4 +...+ Stirling2(9*n, n)*x^n +...
where
A(x) = 1 + 1^9*x*exp(-1^9*x) + 2^18*exp(-2^9*x)*x^2/2! + 3^27*exp(-3^9*x)*x^3/3! + 4^36*exp(-4^9*x)*x^4/4! + 5^45*exp(-5^9*x)*x^5/5! +...
is a power series in x with integer coefficients.

Mathematica

Table[StirlingS2[9*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, n, (k^9)^k*exp(-k^9*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^9)^k*x^k/(1+k^9*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(8*n))), 8*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(9*n, n)}
for(n=0, 12, print1(a(n), ", "))

Crossrefs

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222528, A222530, A217900.

Sequence in context: A161215 A069392 A289479 * A069278 A190780 A017698

Adjacent sequences: A222526 A222527 A222528 * A222530 A222531 A222532

Keyword

nonn

Author

Paul D. Hanna, Feb 23 2013

Status

approved