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Terms Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

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(PARI) seq(nmax, b)={my(v, n, k, irow, f);

T(n,k)=(k)^(k-1)*(n-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.

There are many binomial decompositions of n^n, some with all terms positive like this one (see A243203). However, for every n, the terms corresponding to k=1..n in this one are exceptionally similar in value (at least on log scale).

Stanislav Sykora, <a href="/A244137/b244137.txt">Table of n, a(n) for n = rows 0..5150100</a>

S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.004">An Abel's Identity and its Corollaries</a>, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(13), with b=-1.

(PARI) seq(nmax, b)={my(v, n, k, irow, f);

v = vector((nmax+1)*(nmax+2)/2); v[1]=1;

for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;

for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(n+k*b)^(n-k)*binomial(n, k); ); );

return(v); }

a=seq(100, -1);

Cf. A243203, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244138, A244139, A244140, A244141, A244142, A244143.

allocated for Stanislav SykoraTerms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

1, 0, 1, 0, 2, 2, 0, 12, 6, 9, 0, 108, 48, 36, 64, 0, 1280, 540, 360, 320, 625, 0, 18750, 7680, 4860, 3840, 3750, 7776, 0, 326592, 131250, 80640, 60480, 52500, 54432, 117649, 0, 6588344, 2612736, 1575000, 1146880, 945000, 870912, 941192, 2097152

0,5

Stanislav Sykora, <a href="/A244137/b244137.txt">Table of n, a(n) for n = 0..5150</a>

First rows of the triangle, all summing up to n^n:

1,

0, 1,

0, 2, 2,

0, 12, 6, 9,

0, 108, 48, 36, 64,

0, 1280, 540, 360, 320, 625,

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nonn,tabl

Stanislav Sykora, Jun 22 2014

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allocated for Stanislav Sykora

allocated

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