A229234 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n*k*x).

1, 1, 3, 19, 189, 2671, 50253, 1203679, 35548509, 1263153631, 52973381853, 2581493517439, 144317666200029, 9156299509121311, 653254398215833053, 51995430120141924799, 4585316010326597014749, 445304380297565009962591, 47368550666889620425580253, 5492643630110295899167573759

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..300

Formula

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

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

Example

O.g.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 189*x^4 + 2671*x^5 + 50253*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2*1*x)*(1-2*2*x)) + 3!*x^3/((1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4!*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 + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2671*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2/2^2 + (exp(3*x)-1)^3/3^3 + (exp(4*x)-1)^4/4^4 + (exp(5*x)-1)^5/5^5 +...

Mathematica

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

Crossrefs

Cf. A229233, A220181, A108459, A122399.

Cf. A229258, A229259, A229260, A229261.

Sequence in context: A326553 A052888 A141623 * A090354 A119394 A101481

Adjacent sequences: A229231 A229232 A229233 * A229235 A229236 A229237

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2013

Status

approved