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A244133 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).

%I A244133 #8 Jun 25 2014 09:41:20
%S A244133 0,0,1,0,0,2,0,0,-6,9,0,0,12,-72,64,0,0,-20,360,-960,625,0,0,30,-1440,
%T A244133 8640,-15000,7776,0,0,-42,5040,-60480,210000,-272160,117649,0,0,56,
%U A244133 -16128,362880,-2240000,5443200,-5647152,2097152,0,0,-72,48384,-1959552,20160000,-81648000,152473104,-132120576,43046721
%N A244133 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k).
%C A244133 T(n,k)=(k)^(k-1)*(1-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0 by convention.
%H A244133 Stanislav Sykora, <a href="/A244133/b244133.txt">Table of n, a(n) for rows 0..100</a>
%H A244133 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 A244133 First rows of the triangle, all summing up to n:
%e A244133 0,
%e A244133 0, 1,
%e A244133 0, 0, 2,
%e A244133 0, 0, -6, 9,
%e A244133 0, 0, 12, -72, 64,
%e A244133 0, 0, -20, 360, -960, 625,
%o A244133 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244133 v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244133 for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244133   for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k)*binomial(n, k); ); );
%o A244133 return(v); }
%o A244133 a=seq(100,-1);
%Y A244133 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244133 sign,tabl
%O A244133 0,6
%A A244133 _Stanislav Sykora_, Jun 22 2014