A229258 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2*k*x).

1, 1, 3, 31, 573, 18031, 854613, 57433951, 5242645173, 625589806831, 95051257799973, 17976303383444671, 4153215615930529173, 1154304694449774708751, 380809177225169291456133, 147420687475847638142996191, 66303807316628093952943203573

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..227

Formula

a(n) = Sum_{k=0..n} (k^2)^(n-k) * k! * Stirling2(n, k).

E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n^(2*n).

Example

O.g.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 573*x^4 + 18031*x^5 + 854613*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 573*x^4/4! + 18031*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/4^2 + (exp(9*x)-1)^3/9^3 + (exp(16*x)-1)^4/16^4 + (exp(25*x)-1)^5/25^5 +...

Mathematica

Flatten[{1, Table[Sum[(k^2)^(n-k) * k! * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1-m^2*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m^2*x+x*O(x^n))-1)^m/m^(2*m)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, (k^2)^(n-k) * k! * stirling(n, k, 2))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A229257, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Sequence in context: A261471 A273378 A266487 * A319896 A222895 A105293

Adjacent sequences: A229255 A229256 A229257 * A229259 A229260 A229261

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2013

Status

approved