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A244142 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 A244142 #7 Jun 25 2014 09:43:20
%S A244142 0,0,1,0,0,2,0,0,6,-15,0,0,18,-75,196,0,0,54,-375,1372,-3645,0,0,162,
%T A244142 -1875,9604,-32805,87846,0,0,486,-9375,67228,-295245,966306,-2599051,
%U A244142 0,0,1458,-46875,470596,-2657205,10629366,-33787663,91125000
%N A244142 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 A244142 T(n,k)=(-1)^k*k*(2*k-1)^(n-2) for k>1, while T(n,0)=0 and T(1,1)=0^(n-1) by convention.
%H A244142 Stanislav Sykora, <a href="/A244142/b244142.txt">Table of n, a(n) for rows 0..100</a>
%H A244142 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.(21), with a=1, b=2.
%e A244142 The first rows of the triangle are:
%e A244142 0,
%e A244142 0, 1,
%e A244142 0, 0, 2,
%e A244142 0, 0, 6, -15,
%e A244142 0, 0, 18, -75, 196,
%e A244142 0, 0, 54, -375, 1372, -3645
%o A244142 (PARI) seq(nmax)={my(v, n, k, irow);
%o A244142 v = vector((nmax+1)*(nmax+2)/2); v[1]=0;
%o A244142 for(n=1, nmax, irow=1+n*(n+1)/2;
%o A244142   v[irow]=0; if(n==1, v[irow+1]=1, v[irow+1]=0);
%o A244142 for(k=2,n,v[irow+k]=(-1)^k*k*(2*k-1)^(n-2); ); );
%o A244142 return(v); }
%o A244142 a=seq(100);
%Y A244142 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244143.
%K A244142 sign,tabl
%O A244142 0,6
%A A244142 _Stanislav Sykora_, Jun 23 2014