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A244119 Triangle read by rows: terms of a binomial decomposition of 1 as Sum(k=0..n)T(n,k). 29


%S 1,0,1,0,-2,3,0,3,-18,16,0,-4,72,-192,125,0,5,-240,1440,-2500,1296,0,

%T -6,720,-8640,30000,-38880,16807,0,7,-2016,45360,-280000,680400,

%U -705894,262144,0,-8,5376,-217728,2240000,-9072000,16941456,-14680064,4782969

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

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

%C Sequence A161628, arising from a different context, appears to be the same, but with opposite signs of odd rows.

%H Stanislav Sykora, <a href="/A244119/b244119.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.(4) with b=-1.

%e First rows of the triangle, all summing up to 1:

%e 1

%e 0 1

%e 0 -2 3

%e 0 3 -18 16

%e 0 -4 72 -192 125

%e 0 5 -240 1440 -2500 1296

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

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

%o for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;

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

%o );return(v);}

%o a=seq(100,-1);

%Y Cf. A161628, A244116, A244117, A244118, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.

%K sign,tabl

%O 0,5

%A _Stanislav Sykora_, Jun 21 2014

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)