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proposed

Though the origins original contexts are different, this triangle matches that of A066324 except for row 0, and for the zero term of each row. It is conjectured that the present form might be more amenable for On this point, see the construction of generating functionscomment in A243202.

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editing

proposed

Flattened triangle of the terms Terms of an a particular integer decomposition of N^N.

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. The sum of any row is N^N. Note that all rows start with 0, which makes them easily recognizable.

The sum of every row is N^N.

Though the contexts origins are very different, this triangle matches that of A066324 except for row 0, and for the zero term of each row. It is conjectured that the present form might be more amenable for the construction of generating functions.

Stanislav Sykora, <a href="/A243203/b243203.txt">Table of n, a(n) for n = rows 0..5150100, flattened</a>

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.

allocated for Stanislav SykoraFlattened triangle of the terms of an integer decomposition of N^N.

0, 0, 1, 0, 2, 2, 0, 9, 12, 6, 0, 64, 96, 72, 24, 0, 625, 1000, 900, 480, 120, 0, 7776, 12960, 12960, 8640, 3600, 720, 0, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 0

0,5

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. The sum of any row is N^N. Note that all rows start with 0, which makes them easily recognizable.

Though the contexts are very different, this triangle matches that of A066324 except for row 0, and for the zero term of each row.

Stanislav Sykora, <a href="/A243203/b243203.txt">Table of n, a(n) for n = 0..5150</a>

c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.

The first rows of the triangle are (first item is the row number N):

0 0

1 0, 1

2 0, 2, 2

3 0, 9, 12, 6

4 0, 64, 96, 72, 24

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

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

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

8 0, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320

(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)*binomial(n, j); ); );

return(v); }

Cf. A066324, A243202.

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nonn,tabl

Stanislav Sykora, Jun 01 2014

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editing

allocated for Stanislav Sykora

allocated

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