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

0,5

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.

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, 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

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

Crossrefs

Cf. A066324, A243202.

Sequence in context: A179198 A372390 A117739 * A268652 A111810 A019265

Adjacent sequences: A243200 A243201 A243202 * A243204 A243205 A243206

Keyword

nonn,tabl

Author

Stanislav Sykora, Jun 01 2014

Status

approved