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A244131 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

%I A244131 #9 Jun 25 2014 09:40:58
%S A244131 0,0,1,0,4,-2,0,12,-18,9,0,32,-108,144,-64,0,80,-540,1440,-1600,625,0,
%T A244131 192,-2430,11520,-24000,22500,-7776,0,448,-10206,80640,-280000,472500,
%U A244131 -381024,117649,0,1024,-40824,516096,-2800000,7560000,-10668672,7529536,-2097152
%N A244131 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).
%C A244131 T(n,k)=(-k)^(k-1)*(1+k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0 by convention.
%H A244131 Stanislav Sykora, <a href="/A244131/b244131.txt">Table of n, a(n) for rows 0..100</a>
%H A244131 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 A244131 First rows of the triangle, all summing up to n:
%e A244131 0,
%e A244131 0, 1,
%e A244131 0, 4, -2,
%e A244131 0, 12, -18, 9,
%e A244131 0, 32, -108, 144, -64,
%e A244131 0, 80, -540, 1440, -1600, 625,
%o A244131 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244131 v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244131 for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244131   for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n,k); ); );
%o A244131 return(v); }
%o A244131 a=seq(100,1);
%Y A244131 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244131 sign,tabl
%O A244131 0,5
%A A244131 _Stanislav Sykora_, Jun 22 2014