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A244130 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum_{k=0..n} T(n,k)*binomial(n,k).

%I A244130 #11 Dec 29 2023 10:40:13
%S A244130 0,0,1,0,2,-2,0,4,-6,9,0,8,-18,36,-64,0,16,-54,144,-320,625,0,32,-162,
%T A244130 576,-1600,3750,-7776,0,64,-486,2304,-8000,22500,-54432,117649,0,128,
%U A244130 -1458,9216,-40000,135000,-381024,941192,-2097152,0,256,-4374,36864,-200000,810000,-2667168,7529536,-18874368,43046721
%N A244130 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n as Sum_{k=0..n} T(n,k)*binomial(n,k).
%C A244130 T(n,k) = (-k)^(k-1)*(1+k)^(n-k) for k>0, while T(n,0)=0 by convention.
%H A244130 Stanislav Sykora, <a href="/A244130/b244130.txt">Table of n, a(n) for rows 0..100</a>
%H A244130 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.(12), with b=1.
%e A244130 The first rows of the triangle are:
%e A244130 0,
%e A244130 0, 1,
%e A244130 0, 2, -2,
%e A244130 0, 4, -6, 9,
%e A244130 0, 8, -18, 36, -64,
%e A244130 0, 16, -54, 144, -320, 625,
%o A244130 (PARI) seq(nmax,b)={my(v,n,k,irow);
%o A244130 v = vector((nmax+1)*(nmax+2)/2);v[1]=0;
%o A244130 for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
%o A244130   for(k=1,n,v[irow+k]=(-k*b)^(k-1)*(1+k*b)^(n-k);););
%o A244130 return(v);}
%o A244130 a=seq(100,1);
%Y A244130 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244130 sign,tabl
%O A244130 0,5
%A A244130 _Stanislav Sykora_, Jun 22 2014