A243202 Coefficients of a particular decomposition of N^N in terms of binomial coefficients.

0, 0, 1, 0, 1, 2, 0, 3, 4, 6, 0, 16, 16, 18, 24, 0, 125, 100, 90, 96, 120, 0, 1296, 864, 648, 576, 600, 720, 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040, 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320, 0

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

Cf. A243203.

Sequence in context: A139637 A307734 A363346 * A257136 A344059 A258871

Adjacent sequences: A243199 A243200 A243201 * A243203 A243204 A243205

Keyword

nonn,easy,tabl

Author

Stanislav Sykora, Jun 01 2014

Status

approved