A229233 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n*k*x).

1, 1, 2, 8, 48, 387, 4043, 52425, 819346, 15133184, 324769270, 7986143453, 222514878501, 6958782341565, 242274294115558, 9324382604206368, 394282071192289024, 18218582054356563951, 915480348188869318723, 49812603754178905560085, 2923492374797360684715882

Offset

0,3

Comments

Compare to an o.g.f. of Bell numbers (A000110): Sum_{n>=0} x^n/Product_{k=1..n} (1-k*x).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..332

Formula

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

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

Example

O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 48*x^4 + 387*x^5 + 4043*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2*1*x)*(1-2*2*x)) + x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + x^4/((1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 48*x^4/4! + 387*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/(2!*2^2) + (exp(3*x)-1)^3/(3!*3^3) + (exp(4*x)-1)^4/(4!*4^4) + (exp(5*x)-1)^5/(5!*5^5) +...

Mathematica

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

Crossrefs

Cf. A229234, A220181, A108459, A122399.

Cf. A229258, A229259, A229260, A229261.

Sequence in context: A009812 A232005 A351422 * A063075 A177386 A112541

Adjacent sequences: A229230 A229231 A229232 * A229234 A229235 A229236

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2013

Status

approved