A244141 - OEIS

%I #7 Jun 25 2014 09:43:09

%S 0,0,-1,0,0,2,0,0,0,-3,0,0,0,-12,16,0,0,0,-30,160,-135,0,0,0,-60,960,

%T -2430,1536,0,0,0,-105,4480,-25515,43008,-21875,0,0,0,-168,17920,

%U -204120,688128,-875000,373248,0,0,0,-252,64512,-1377810,8257536,-19687500,20155392,-7411887

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

%C T(n,k)=(-1)^k*k*(k-2)^(n-2)*binomial(n,k) for k>1, while T(n,0)=0 and T(1,1)=-0^(n-1) by convention.

%H Stanislav Sykora, <a href="/A244141/b244141.txt">Table of n, a(n) for rows 0..100</a>

%H 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=2, b=1.

%e First rows of the triangle, all summing up to n*(-1)^n:

%e 0,

%e 0, -1,

%e 0, 0, 2,

%e 0, 0, 0, -3,

%e 0, 0, 0, -12, 16,

%e 0, 0, 0, -30, 160, -135,

%e 0, 0, 0, -60, 960, -2430, 1536,

%o (PARI) seq(nmax)={my(v, n, k, irow);

%o v = vector((nmax+1)*(nmax+2)/2); v[1]=0;

%o for(n=1, nmax, irow=1+n*(n+1)/2;

%o   v[irow]=0; if(n==1, v[irow+1]=-1, v[irow+1]=0);

%o for(k=2, n, v[irow+k]=(-1)^k*k*(k-2)^(n-2)*binomial(n,k); ); );

%o return(v); }

%o a=seq(100);

%Y 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, A244142, A244143.

%K sign,tabl

%O 0,6

%A _Stanislav Sykora_, Jun 23 2014