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A244122 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k)*binomial(n,k).

%I A244122 #12 Jun 25 2014 09:38:45
%S A244122 1,0,1,0,-2,8,0,3,-30,108,0,-4,96,-588,2048,0,5,-280,2880,-14580,
%T A244122 50000,0,-6,768,-13122,96000,-439230,1492992,0,7,-2016,56700,-596288,
%U A244122 3628800,-15594306,52706752,0,-8,5120,-235224,3538944,-28561000,154893312,-637875000,2147483648,0
%N A244122 Triangle read by rows: coefficients T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k)*binomial(n,k).
%C A244122 T(n,k)=n*(n+k)^(k-1)*(-k)^(n-k) for k>0, while T(n,0)=0^n by convention.
%H A244122 Stanislav Sykora, <a href="/A244122/b244122.txt">Table of n, a(n) for rows 0..100</a>
%H A244122 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.(5), with b=-1.
%e A244122 The first rows of the triangle are:
%e A244122 1
%e A244122 0  1
%e A244122 0 -2    8
%e A244122 0  3  -30  108
%e A244122 0 -4   96 -588   2048
%e A244122 0  5 -280 2880 -14580 50000
%o A244122 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244122   v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
%o A244122   for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244122   for(k=1, n, v[irow+k] = n*(n-k*b)^(k-1)*(k*b)^(n-k); ); );
%o A244122   return(v); }
%o A244122   a=seq(100,-1);
%Y A244122 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244122 sign,tabl
%O A244122 0,5
%A A244122 _Stanislav Sykora_, Jun 21 2014