A229259 O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2*k*x).

1, 1, 9, 259, 15789, 1693771, 287145789, 71487432619, 24798142070109, 11518873418467051, 6945333793188487869, 5301472723402989073579, 5018547949600497090304029, 5790959348524892656227425131, 8026963462960378548022418765949, 13197920271743736945902641688868139

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..201

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^n.

Example

O.g.f.: A(x) = 1 + x + 9*x^2 + 259*x^3 + 15789*x^4 + 1693771*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^2*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^3*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^4*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 + 9*x^2/2! + 259*x^3/3! + 15789*x^4/4! + 1693771*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2^2 + (exp(9*x)-1)^3/3^3 + (exp(16*x)-1)^4/4^4 + (exp(25*x)-1)^5/5^5 +...

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!*m^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^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, A229258, A229260, A229261, A229233, A229234, A220181, A122399.

Sequence in context: A157575 A072158 A001823 * A376096 A117796 A117051

Adjacent sequences: A229256 A229257 A229258 * A229260 A229261 A229262

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2013

Status

approved