|
%I #12 Jun 02 2014 01:45:29
%S 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,
%T 576,600,720,0,16807,9604,6174,4704,4200,4320,5040,0,262144,131072,
%U 73728,49152,38400,34560,35280,40320,0
%N Coefficients of a particular decomposition of N^N in terms of binomial coefficients.
%C 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.
%H Stanislav Sykora, <a href="/A243202/b243202.txt">Table of n, a(n) for rows 0..100, flattened</a>
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">A Random Mapping Statistics and a Related Identity</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, June 2014.
%F c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.
%e The first rows of the triangle are (first item is the row number N):
%e 0 0
%e 1 0, 1
%e 2 0, 1, 2
%e 3 0, 3, 4, 6
%e 4 0, 16, 16, 18, 24
%e 5 0, 125, 100, 90, 96, 120
%e 6 0, 1296, 864, 648, 576, 600, 720
%e 7 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040
%e 8 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320
%o (PARI) A243202(maxrow) = {
%o my(v,n,j,irow,f);v = vector((maxrow+1)*(maxrow+2)/2);
%o for(n=1,maxrow,irow=1+n*(n+1)/2;v[irow]=0;f=1;
%o for(j=1,n,f *= j;v[irow+j] = j*f*n^(n-j-1);););
%o return(v);}
%Y Cf. A243203.
%K nonn,easy,tabl
%O 0,6
%A _Stanislav Sykora_, Jun 01 2014
|
