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A244139 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(n-1) as Sum(k=0..n)T(n,k).

%I A244139 #8 Jun 25 2014 09:42:54
%S A244139 0,0,0,0,0,2,0,0,12,-6,0,0,48,-72,36,0,0,160,-540,720,-320,0,0,480,
%T A244139 -3240,8640,-9600,3750,0,0,1344,-17010,80640,-168000,157500,-54432,0,
%U A244139 0,3584,-81648,645120,-2240000,3780000,-3048192,941192,0,0,9216,-367416,4644864,-25200000,68040000,-96018048,67765824,-18874368
%N A244139 Triangle read by rows: terms T(n,k) of a binomial decomposition of n*(n-1) as Sum(k=0..n)T(n,k).
%C A244139 T(n,k)=k*(1-k)^(k-2)*k^(n-k)*binomial(n,k) for k>1, while T(n,0)=T(n,1)=0 by convention.
%H A244139 Stanislav Sykora, <a href="/A244139/b244139.txt">Table of n, a(n) for rows 0..100</a>
%H A244139 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.(19), with a=1.
%e A244139 First rows of the triangle, all summing up to n*(n-1):
%e A244139 0,
%e A244139 0, 0,
%e A244139 0, 0, 2,
%e A244139 0, 0, 12, -6,
%e A244139 0, 0, 48, -72, 36,
%e A244139 0, 0, 160, -540, 720, -320,
%e A244139 0, 0, 480, -3240, 8640, -9600, 3750,
%o A244139 (PARI) seq(nmax)={my(v, n, k, irow);
%o A244139 v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244139 for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0; v[irow+1]=0;
%o A244139   for(k=2, n, v[irow+k]=k*(1-k)^(k-2)*k^(n-k)*binomial(n,k); ); );
%o A244139 return(v); }
%o A244139 a=seq(100);
%Y A244139 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244140, A244141, A244142, A244143.
%K A244139 sign,tabl
%O A244139 0,6
%A A244139 _Stanislav Sykora_, Jun 22 2014