0,3

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..410 (first 181 terms from Alois P. Heinz)

R. K. Guy, Letter to N. J. A. Sloane, Jun 21, 1975

a(n) = n! * [x^n] (exp(x)-x)^n.

a(n) ~ (1-exp(-1))^(n+1/2) * n^n. - Vaclav Kotesovec, Jul 23 2014

E.g.f.: 1 / ((1 + x) * (1 + LambertW(-x/(1 + x)))). - Ilya Gutkovskiy, Jan 25 2020

a(n) = Sum_{k=0..n} (-1)^(n-k)*(n-k)!*k^k*binomial(n,k)^2. - Ridouane Oudra, Jul 14 2025

a(2) = 2: (1,1), (2,2).

a(3) = 3: (1,1,1), (2,2,2), (3,3,3).

with(combinat):

b:= proc(t, i, u) option remember; `if`(t=0, 1,

`if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)

*b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))

end:

a:= n-> b(n$3):

seq(a(n), n=0..25);

multinomial[n_, k_List] := n!/Times @@ (k!); b[t_, i_, u_] := b[t, i, u] = If[t == 0, 1, If[i < 2, 0, b[t, i - 1, u] + Sum[multinomial[t, Join[{ t - i*j}, Array[i &, j]]] * b[t - i*j, i - 1, u - j]*u!/(u - j)!/j!, {j, 1, t/i}]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Table[n!*SeriesCoefficient[(E^x-x)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 23 2014 *)

Flatten[{1, Table[(-1)^n*n! + Sum[Binomial[n, j]^2*(-1)^j*(n-j)^(n-j)*j!, {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 25 2014 *)

(PARI) {a(n) = n! * polcoeff((exp(x + x*O(x^n)) - x)^n, n)}

for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jan 30 2015

Column k=0 of A206823.

A diagonal of A241580. Cf. also A241581.

Column k=1 of A245405.

Cf. A245496.

Sequence in context: A375695 A355843 A097170 * A355235 A349560 A176366

Adjacent sequences: A231794 A231795 A231796 * A231798 A231799 A231800

Alois P. Heinz, Nov 13 2013

approved