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 A243203 Terms of a particular integer decomposition of N^N. 4

%I

%S 0,0,1,0,2,2,0,9,12,6,0,64,96,72,24,0,625,1000,900,480,120,0,7776,

%T 12960,12960,8640,3600,720,0,117649,201684,216090,164640,88200,30240,

%U 5040,0,2097152,3670016,4128768,3440640,2150400,967680,282240,40320,0

%N Terms of a particular integer decomposition of N^N.

%C 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.

%C The sum of every row is N^N.

%C 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.

%H Stanislav Sykora, <a href="/A243203/b243203.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, 2, 2

%e 3 0, 9, 12, 6

%e 4 0, 64, 96, 72, 24

%e 5 0, 625, 1000, 900, 480, 120

%e 6 0, 7776, 12960, 12960, 8640, 3600, 720

%e 7 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040

%e 8 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 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)*binomial(n,j);););

%o return(v);}

%Y Cf. A066324, A243202.

%K nonn,tabl

%O 0,5

%A _Stanislav Sykora_, Jun 01 2014

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Last modified May 15 07:17 EDT 2021. Contains 343909 sequences. (Running on oeis4.)