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Coefficients Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k)*binomial(n,k).

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T(n,k)=(-1)^k*k*(2*k-1)^(n-2) for k>1, while T(n,0)=0 and T(1,1)=0^(n-1) by convention.

Stanislav Sykora, <a href="/A244142/b244142.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.(21), with a=1, b=2.

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

0, 0, 1, 0, 0, 2, 0, 0, 6, -15, 0, 0, 18, -75, 196, 0, 0, 54, -375, 1372, -3645, 0, 0, 162, -1875, 9604, -32805, 87846, 0, 0, 486, -9375, 67228, -295245, 966306, -2599051, 0, 0, 1458, -46875, 470596, -2657205, 10629366, -33787663, 91125000

0,6

Stanislav Sykora, <a href="/A244142/b244142.txt">Table of n, a(n) for n = 0..5150</a>

The first rows of the triangle are:

0,

0, 1,

0, 0, 2,

0, 0, 6, -15,

0, 0, 18, -75, 196,

0, 0, 54, -375, 1372, -3645

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

v = vector((nmax+1)*(nmax+2)/2); v[1]=0;

for(n=1, nmax, irow=1+n*(n+1)/2;

v[irow]=0; if(n==1, v[irow+1]=1, v[irow+1]=0);

for(k=2, n, v[irow+k]=(-1)^k*k*(2*k-1)^(n-2); ); );

return(v); }

a=seq(100);

Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244143.

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Stanislav Sykora, Jun 23 2014

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allocated for Stanislav Sykora

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