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A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).

%I A244128 #13 Jun 24 2021 09:29:03
%S A244128 0,1,0,1,-2,0,1,-4,9,0,1,-8,27,-64,0,1,-16,81,-256,625,0,1,-32,243,
%T A244128 -1024,3125,-7776,0,1,-64,729,-4096,15625,-46656,117649,0,1,-128,2187,
%U A244128 -16384,78125,-279936,823543,-2097152,0,1,-256,6561,-65536,390625,-1679616,5764801,-16777216,43046721
%N A244128 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of 0^(n-1) as Sum(k=0..n)T(n,k)*binomial(n,k).
%C A244128 T(n,k)=(-k)^(k-1)*k^(n-k) for k>0, while T(n,0)=0 by convention. The flattened triangle start with row 1, coefficient T(1,0).
%C A244128 Resembles A076014, but with added powers of 0, and with sign-alternating columns.
%H A244128 Stanislav Sykora, <a href="/A244128/b244128.txt">Table of n, a(n) for rows 1..100</a>
%H A244128 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.(11), with b=1.
%e A244128 The first rows of the triangle (starting at n=1):
%e A244128 0, 1,
%e A244128 0, 1, -2,
%e A244128 0, 1, -4, 9,
%e A244128 0, 1, -8, 27, -64,
%e A244128 0, 1, -16, 81, -256, 625,
%e A244128 0, 1, -32, 243, -1024, 3125, -7776,
%o A244128 (PARI) seq(nmax,b)={my(v,n,k,irow);
%o A244128 v = vector((nmax+1)*(nmax+2)/2-1);
%o A244128 for(n=1,nmax,irow=n*(n+1)/2;v[irow]=0;
%o A244128   for(k=1,n,v[irow+k]=(-1)^(k-1)*(k*b)^(n-1);););
%o A244128 return(v);}
%o A244128 a=seq(100,1);
%Y A244128 Cf. A076014, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244128 sign,tabf
%O A244128 1,5
%A A244128 _Stanislav Sykora_, Jun 22 2014