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A244132 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).

%I A244132 #8 Jun 25 2014 09:41:07
%S A244132 0,0,1,0,0,2,0,0,-2,9,0,0,2,-18,64,0,0,-2,36,-192,625,0,0,2,-72,576,
%T A244132 -2500,7776,0,0,-2,144,-1728,10000,-38880,117649,0,0,2,-288,5184,
%U A244132 -40000,194400,-705894,2097152,0,0,-2,576,-15552,160000,-972000,4235364,-14680064,43046721
%N A244132 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).
%C A244132 T(n,k)=(k)^(k-1)*(1-k)^(n-k) for k>0, while T(n,0)=0 by convention.
%H A244132 Stanislav Sykora, <a href="/A244132/b244132.txt">Table of n, a(n) for rows 0..100</a>
%H A244132 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.(12), with b=-1.
%e A244132 The first rows of the triangle are:
%e A244132 0,
%e A244132 0, 1,
%e A244132 0, 0, 2,
%e A244132 0, 0, -2, 9,
%e A244132 0, 0, 2, -18, 64,
%e A244132 0, 0, -2, 36, -192, 625,
%o A244132 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244132 v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244132 for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244132   for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k); ); );
%o A244132 return(v); }
%o A244132 a=seq(100,-1);
%Y A244132 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244132 sign,tabl
%O A244132 0,6
%A A244132 _Stanislav Sykora_, Jun 22 2014