A244118 - OEIS

%I #18 Jun 17 2022 14:19:35

%S 1,0,1,0,-1,3,0,1,-6,16,0,-1,12,-48,125,0,1,-24,144,-500,1296,0,-1,48,

%T -432,2000,-6480,16807,0,1,-96,1296,-8000,32400,-100842,262144,0,-1,

%U 192,-3888,32000,-162000,605052,-1835008,4782969,0,1,-384,11664,-128000,810000,-3630312,12845056,-38263752,100000000

%N Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 1 as Sum_{k=0..n} T(n,k)*binomial(n,k).

%C T(n,k) = (1+k)^(k-1)*(-k)^(n-k) for k>0, where T(n,0) = 0^n.

%H Stanislav Sykora, <a href="/A244118/b244118.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 The first rows of the triangle are:

%e 1

%e 0  1

%e 0 -1   3

%e 0  1  -6  16

%e 0 -1  12 -48  125

%e 0  1 -24 144 -500 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););

%o   );return(v);}

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

%Y Cf. A244116, A244117, A244119, 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 0,6

%A _Stanislav Sykora_, Jun 21 2014