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A244143 Triangle read by rows: terms T(n,k) of a binomial decomposition of n as Sum(k=0..n)T(n,k). 28


%S 0,0,1,0,0,2,0,0,18,-15,0,0,108,-300,196,0,0,540,-3750,6860,-3645,0,0,

%T 2430,-37500,144060,-196830,87846,0,0,10206,-328125,2352980,-6200145,

%U 6764142,-2599051,0,0,40824,-2625000,32941720,-148803480,297622248,-270301304,91125000

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

%C T(n,k)=(-1)^k*k*(2*k-1)^(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="/A244143/b244143.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=1, b=2.

%e First rows of the triangle, all summing up to n:

%e 0,

%e 0, 1,

%e 0, 0, 2,

%e 0, 0, 18, -15,

%e 0, 0, 108, -300, 196,

%e 0, 0, 540, -3750, 6860, -3645,

%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*(2*k-1)^(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, A244141, A244142.

%K sign,tabl

%O 0,6

%A _Stanislav Sykora_, Jun 23 2014

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Last modified November 16 19:17 EST 2019. Contains 329201 sequences. (Running on oeis4.)