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%I #10 Jun 25 2014 09:40:07
%S 0,0,1,0,0,3,0,0,-9,16,0,0,18,-128,125,0,0,-30,640,-1875,1296,0,0,45,
%T -2560,16875,-31104,16807,0,0,-63,8960,-118125,435456,-588245,262144,
%U 0,0,84,-28672,708750,-4644864,11764900,-12582912,4782969
%N Triangle read by rows: terms T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k).
%C T(n,k)=(1+k)^(k-1)*(1-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0 by convention.
%H Stanislav Sykora, <a href="/A244127/b244127.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 First rows of the triangle, all summing up to 2^n-1:
%e 0,
%e 0, 1,
%e 0, 0, 3,
%e 0, 0, -9, 16,
%e 0, 0, 18, -128, 125,
%e 0, 0, -30, 640, -1875, 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)*binomial(n, k); ); );
%o return(v); }
%o a=seq(100,-1)
%Y Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, 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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