|
%I #11 Jun 25 2014 09:39:20
%S 0,0,1,0,2,-1,0,4,-3,4,0,8,-9,16,-27,0,16,-27,64,-135,256,0,32,-81,
%T 256,-675,1536,-3125,0,64,-243,1024,-3375,9216,-21875,46656,0,128,
%U -729,4096,-16875,55296,-153125,373248,-823543,0,256,-2187,16384,-84375,331776,-1071875,2985984,-7411887,16777216
%N Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 2^n-1 as Sum(k=0..n)T(n,k)*binomial(n,k).
%C T(n,k)=(1-k)^(k-1)*(1+k)^(n-k) for k>0, while T(n,0)=0 by convention.
%H Stanislav Sykora, <a href="/A244124/b244124.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.(6), with b=1 and a=1.
%e The first rows of the triangle are:
%e 0
%e 0 1
%e 0 2 -1
%e 0 4 -3 4
%e 0 8 -9 16 -27
%e 0 16 -27 64 -135 256
%o (PARI) seq(nmax,b)={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;v[irow]=0;
%o for(k=1,n,v[irow+k]=(1-k*b)^(k-1)*(1+k*b)^(n-k);););
%o return(v);}
%o a=seq(100,1)
%Y Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, 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
|
