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A244125 Triangle read by rows: terms T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k).

%I A244125 #12 Jun 25 2014 09:39:35
%S A244125 0,0,1,0,4,-1,0,12,-9,4,0,32,-54,64,-27,0,80,-270,640,-675,256,0,192,
%T A244125 -1215,5120,-10125,9216,-3125,0,448,-5103,35840,-118125,193536,
%U A244125 -153125,46656,0,1024,-20412,229376,-1181250,3096576,-4287500,2985984,-823543
%N A244125 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 A244125 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 A244125 Stanislav Sykora, <a href="/A244125/b244125.txt">Table of n, a(n) for rows 0..100</a>
%H A244125 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 A244125 First rows of the triangle, all summing up to 2^n-1:
%e A244125 1
%e A244125 0 1
%e A244125 0 4  -1
%e A244125 0 12 -9 4
%e A244125 0 32 -54 64 -27
%e A244125 0 80 -270 640 -675 256
%o A244125 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244125   v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244125   for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244125   for(k=1, n, v[irow+k]=(1-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n,k); ); );
%o A244125   return(v); }
%o A244125   a=seq(100, 1)
%Y A244125 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244125 sign,tabl
%O A244125 0,5
%A A244125 _Stanislav Sykora_, Jun 21 2014