OFFSET
0,3
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..410 (first 180 terms from Alois P. Heinz)
R. K. Guy, Letter to N. J. A. Sloane, Jun 21, 1975
FORMULA
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
EXAMPLE
a(2) = 2: (1,1), (2,2).
a(3) = 3: (1,1,1), (2,2,2), (3,3,3).
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 13 2013
STATUS
approved