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A244116 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).

%I A244116 #31 Jan 28 2023 09:26:59
%S A244116 1,0,1,0,1,-1,0,1,-2,4,0,1,-4,12,-27,0,1,-8,36,-108,256,0,1,-16,108,
%T A244116 -432,1280,-3125,0,1,-32,324,-1728,6400,-18750,46656,0,1,-64,972,
%U A244116 -6912,32000,-112500,326592,-823543,0,1,-128,2916,-27648,160000,-675000,2286144,-6588344,16777216
%N A244116 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 A244116 T(n,k) = (1-k)^(k-1) * k^(n-k) for k>0, and T(n,0) = 0^n by convention.
%H A244116 Stanislav Sykora, <a href="/A244116/b244116.txt">Table of n, rows 0..100</a>
%H A244116 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. See eq.(4) with b=1.
%e A244116 The first few rows of the triangle are:
%e A244116   1
%e A244116   0 1
%e A244116   0 1 -1
%e A244116   0 1 -2 4
%e A244116   0 1 -4 12  -27
%e A244116   0 1 -8 36 -108 256
%e A244116   ...
%p A244116 A244116 := (n, j) -> (-1)^(j + 1) * j^(n - j) * (j - 1)^(j - 1):
%p A244116 for n from 0 to 9 do seq(A244116(n, k), k = 0..n) od; # _Peter Luschny_, Jan 28 2023
%o A244116 (PARI) seq(nmax,b)={my(v,n,k,irow);
%o A244116   v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
%o A244116   for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
%o A244116     for(k=1,n,v[irow+k] = (1-k*b)^(k-1)*(k*b)^(n-k););
%o A244116   );return(v);}
%o A244116   a=seq(100,1);
%Y A244116 Cf. A244117, A244118, 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.
%Y A244116 Cf. A357247.
%K A244116 sign,tabl
%O A244116 0,9
%A A244116 _Stanislav Sykora_, Jun 21 2014