OFFSET
0,2
COMMENTS
The n-th row of array A089900 is the n-th binomial transform of the factorials found in row 0: {1!,2!,3!,...,(n+1)!,...}. The hyperbinomial transform of this main diagonal gives: {1,4,27,...,(n+1)^(n+1),...}, which is the next lower diagonal in array A089900.
a(n), for n>=1, is the number of colored labeled mappings from n points to themselves, where each component is one of three colors. - Steven Finch, Nov 28 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..380
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 162.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
FORMULA
a(n) = Sum_{k=0..n} n^(n-k) * C(n,k) * (k+1)!.
a(n) = Sum_{k=0..n} -(n-k-1)^(n-k-1) * C(n,k) * (k+1)^(k+1).
E.g.f.: 1 / (1 + LambertW(-x))^3.
E.g.f.: (Sum_{n>=0} (n+1)^(n+1) * x^n/n!) * (Sum_{n>=0} -(n-1)^(n-1) * x^n/n!).
a(n) ~ n^(n+1) * (1 + sqrt(Pi/(2*n))). - Vaclav Kotesovec, Jul 09 2013
a(n) = (n^(n + 2) + exp(n)*Gamma(n + 2, n)) / (n + 1). - Peter Luschny, Nov 29 2021
MATHEMATICA
CoefficientList[Series[1/(1+LambertW[-x])^3, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Flatten[{1, Table[Sum[n^(n-k)*Binomial[n, k]*(k+1)!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 09 2013 *)
a[n_] := (n^(n + 2) + Exp[n] Gamma[n + 2, n]) / (n + 1);
Table[a[n], {n, 0, 17}] (* Peter Luschny, Nov 29 2021 *)
PROG
(PARI) /* As (n+1)-th term of the n-th binomial transform of {(n+1)!}: */
{a(n)=if(n<0, 0, sum(i=0, n, n^(n-i)*binomial(n, i)*(i+1)!))}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As (n+1)-th term of inverse hyperbinomial of {(n+1)^(n+1)}: */
{a(n)=if(n<0, 0, sum(i=0, n, -(n-i-1)^(n-i-1)*binomial(n, i)*(i+1)^(i+1)))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2003
STATUS
approved