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%I #15 Jun 24 2014 00:40:21
%S 1,0,1,0,2,-1,0,3,-6,4,0,4,-24,48,-27,0,5,-80,360,-540,256,0,6,-240,
%T 2160,-6480,7680,-3125,0,7,-672,11340,-60480,134400,-131250,46656,0,8,
%U -1792,54432,-483840,1792000,-3150000,2612736,-823543,0,9,-4608,244944,-3483648,20160000,-56700000,82301184,-59295096,16777216
%N Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k).
%C T(n,k)=(1-k)^(k-1)*k^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
%H Stanislav Sykora, <a href="/A244117/b244117.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.(4), with b=1.
%e First rows of the triangle, all summing up to 1:
%e 1
%e 0 1
%e 0 2 -1
%e 0 3 -6 4
%e 0 4 -24 48 -27
%e 0 5 -80 360 -540 256
%o (PARI) seq(nmax,b)={my(v,n,k,irow);
%o v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
%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)*(k*b)^(n-k)*binomial(n,k););
%o );return(v);}
%o a=seq(100,1);
%Y Cf. A244116, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K sign,tabl
%O 0,5
%A _Stanislav Sykora_, Jun 21 2014
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