OFFSET
0,6
COMMENTS
a(n) is an element in the triangle of coefficients c(N,j), N = 0,1,2,3,... denoting a row, and j = 0,1,2,...r, specified numerically by the formula below. For any row N, Sum(j=0..N)(c(N,j)*binomial(N,j)) = N^N. Note that all rows start with 0, which makes them easily recognizable. It is believed that keeping the zero terms is preferable because it makes the summation run over all admissible j's in the binomial.
LINKS
Stanislav Sykora, Table of n, a(n) for rows 0..100, flattened
S. Sykora, A Random Mapping Statistics and a Related Identity, Stan's Library, Volume V, June 2014.
FORMULA
c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.
EXAMPLE
The first rows of the triangle are (first item is the row number N):
0 0
1 0, 1
2 0, 1, 2
3 0, 3, 4, 6
4 0, 16, 16, 18, 24
5 0, 125, 100, 90, 96, 120
6 0, 1296, 864, 648, 576, 600, 720
7 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040
8 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320
PROG
(PARI) A243202(maxrow) = {
my(v, n, j, irow, f); v = vector((maxrow+1)*(maxrow+2)/2);
for(n=1, maxrow, irow=1+n*(n+1)/2; v[irow]=0; f=1;
for(j=1, n, f *= j; v[irow+j] = j*f*n^(n-j-1); ); );
return(v); }
CROSSREFS
KEYWORD
AUTHOR
Stanislav Sykora, Jun 01 2014
STATUS
approved