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A244126 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k). 28

%I

%S 0,0,1,0,0,3,0,0,-3,16,0,0,3,-32,125,0,0,-3,64,-375,1296,0,0,3,-128,

%T 1125,-5184,16807,0,0,-3,256,-3375,20736,-84035,262144,0,0,3,-512,

%U 10125,-82944,420175,-1572864,4782969,0,0,-3,1024,-30375,331776,-2100875,9437184,-33480783,100000000,0,0,3,-2048,91125,-1327104,10504375,-56623104,234365481,-800000000,2357947691,0,0,-3,4096,-273375,5308416,-52521875

%N Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).

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

%H Stanislav Sykora, <a href="/A244126/b244126.txt">Table of n, a(n) for rows 0..100</a>

%H 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.(6), with b=-1 and a=1.

%e The first rows of the triangle are:

%e 0,

%e 0, 1,

%e 0, 0, 3,

%e 0, 0, -3, 16,

%e 0, 0, 3, -32, 125,

%e 0, 0, -3, 64, -375, 1296,

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

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

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

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

%o return(v); }

%o a=seq(100,-1)

%Y Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

%K sign,tabl

%O 0,6

%A _Stanislav Sykora_, Jun 21 2014

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Last modified January 20 23:20 EST 2019. Contains 319343 sequences. (Running on oeis4.)