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

Author

Stanislav Sykora, Jun 01 2014

Keywords

• nonn
• easy
• tabl

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.

Examples

			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
		

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. A243203.

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);););
  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.