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

%I A243202 #12 Jun 02 2014 01:45:29
%S A243202 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 A243202 576,600,720,0,16807,9604,6174,4704,4200,4320,5040,0,262144,131072,
%U A243202 73728,49152,38400,34560,35280,40320,0
%N A243202 Coefficients of a particular decomposition of N^N in terms of binomial coefficients.
%C A243202 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 A243202 Stanislav Sykora, <a href="/A243202/b243202.txt">Table of n, a(n) for rows 0..100, flattened</a>
%H A243202 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 A243202 c(N,j)=N^(N-j)*(j/N)*j!  for  N>0 and 0<=j<=N, and c(N,j)=0 otherwise.
%e A243202 The first rows of the triangle are (first item is the row number N):
%e A243202 0 0
%e A243202 1 0, 1
%e A243202 2 0, 1, 2
%e A243202 3 0, 3, 4, 6
%e A243202 4 0, 16, 16, 18, 24
%e A243202 5 0, 125, 100, 90, 96, 120
%e A243202 6 0, 1296, 864, 648, 576, 600, 720
%e A243202 7 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040
%e A243202 8 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320
%o A243202 (PARI) A243202(maxrow) = {
%o A243202   my(v,n,j,irow,f);v = vector((maxrow+1)*(maxrow+2)/2);
%o A243202   for(n=1,maxrow,irow=1+n*(n+1)/2;v[irow]=0;f=1;
%o A243202   for(j=1,n,f *= j;v[irow+j] = j*f*n^(n-j-1);););
%o A243202   return(v);}
%Y A243202 Cf. A243203.
%K A243202 nonn,easy,tabl
%O A243202 0,6
%A A243202 _Stanislav Sykora_, Jun 01 2014