A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8*x) * x^n / n!.

1, 1, 32767, 47063200806, 768305500780164501, 75740854251732106906082250, 31154086963475828638359480518580526, 41929298560838945526242744414099901692285884, 155440114706926165785630654089245708839702615196926765, 1396002062838446082394548660243302585983358463911636390911298400

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..100

Formula

a(n) = Stirling2(8*n, n).

a(n) = [x^(8*n)] (8*n)! * (exp(x) - 1)^n / n!.

a(n) = [x^(7*n)] 1 / Product_{k=1..n} (1-k*x).

a(n) = 1/n! * [x^n] Sum_{k>=0} (k^8)^k*x^k / (1 + k^8*x)^(k+1).

a(n) ~ n^(7*n) * 8^(8*n) / (sqrt(2*Pi*(1-c)*n) * exp(7*n) * (8-c)^(7*n) * c^n), where c = -LambertW(-8*exp(-8)). - Vaclav Kotesovec, May 11 2014

Example

O.g.f.: A(x) = 1 + x + 32767*x^2 + 47063200806*x^3 + 768305500780164501*x^4 +...+ Stirling2(8*n, n)*x^n +...
where
A(x) = 1 + 1^8*x*exp(-1^8*x) + 2^16*exp(-2^8*x)*x^2/2! + 3^24*exp(-3^8*x)*x^3/3! + 4^32*exp(-4^8*x)*x^4/4! + 5^40*exp(-5^8*x)*x^5/5! +...
is a power series in x with integer coefficients.

Mathematica

Table[StirlingS2[8*n, n], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, n, (k^8)^k*exp(-k^8*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^8)^k*x^k/(1+k^8*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(7*n))), 7*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(8*n, n)}
for(n=0, 12, print1(a(n), ", "))

Crossrefs

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222529, A222530, A217900.

Sequence in context: A069390 A069416 A289478 * A232393 A217589 A069276

Adjacent sequences: A222525 A222526 A222527 * A222529 A222530 A222531

Keyword

nonn

Author

Paul D. Hanna, Feb 23 2013

Status

approved