A243203 Terms of a particular integer decomposition of N^N.

0, 0, 1, 0, 2, 2, 0, 9, 12, 6, 0, 64, 96, 72, 24, 0, 625, 1000, 900, 480, 120, 0, 7776, 12960, 12960, 8640, 3600, 720, 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 0

Offset: 0

Author

Stanislav Sykora, Jun 01 2014

Keywords

• nonn
• tabl

Comments

a(n) is an element in the triangle of terms t(N,j) = c(N,j)*binomial(N,j), N = 0,1,2,3,... denoting a row, and j = 0,1,2,...r. The coefficients c(N,j) are specified numerically by the formula below. Note that all rows start with 0, which makes them easily recognizable.

The sum of every row is N^N.

Though the original contexts are different, this triangle matches that of A066324 except for row 0, and for the zero term of each row. On this point, see the comment in A243202.

Examples

			The first rows of the triangle are (first item is the row number N):
0 0
1 0, 1
2 0, 2, 2
3 0, 9, 12, 6
4 0, 64, 96, 72, 24
5 0, 625, 1000, 900, 480, 120
6 0, 7776, 12960, 12960, 8640, 3600, 720
7 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040
8 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320
		

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.

Crossrefs

Cf. A066324, A243202.

Programs

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)*binomial(n,j);););
  return(v);}

Formula

c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.