A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2*k*x).

1, 1, 2, 14, 168, 3147, 90563, 3561231, 185790622, 12599020184, 1071164190670, 111813313594259, 14140296360430353, 2132273568722682621, 378197030144360862958, 78127192632748956075174, 18627308660113953164384812, 5081218748742336002185874439, 1574128413278644602881499193579

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..242

Formula

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

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

Example

O.g.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 168*x^4 + 3147*x^5 + 90563*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2^2*1*x)*(1-2^2*2*x)) + x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 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 + 2*x^2/2! + 14*x^3/3! + 168*x^4/4! + 3147*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/(2!*4^2) + (exp(9*x)-1)^3/(3!*9^3) + (exp(16*x)-1)^4/(4!*16^4) + (exp(25*x)-1)^5/(5!*25^5) +...

Mathematica

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

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m/prod(k=1, m, 1-m^2*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m^2*x+x*O(x^n))-1)^m/(m!*m^(2*m))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, (k^2)^(n-k) * stirling(n, k, 2))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A229258, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Sequence in context: A084946 A047055 A355779 * A188194 A046247 A141012

Adjacent sequences: A229254 A229255 A229256 * A229258 A229259 A229260

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2013

Status

approved