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A244123 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

%I A244123 #11 Jun 25 2014 09:38:55
%S A244123 1,0,1,0,-4,8,0,9,-90,108,0,-16,576,-2352,2048,0,25,-2800,28800,
%T A244123 -72900,50000,0,-36,11520,-262440,1440000,-2635380,1492992,0,49,
%U A244123 -42336,1984500,-20870080,76204800,-109160142,52706752,0,-64,143360,-13172544,247726080,-1599416000,4337012736,-5103000000,2147483648
%N A244123 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
%C A244123 T(n,k)=n*(n+k)^(k-1)*(-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
%H A244123 Stanislav Sykora, <a href="/A244123/b244123.txt">Table of n, a(n) for rows 0..100</a>
%H A244123 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 A244123 First rows of the triangle, all summing up to n^n:
%e A244123 1
%e A244123 0 1
%e A244123 0 -4  8
%e A244123 0, 9 -90 108
%e A244123 0 -16 576 -2352 2048
%e A244123 0, 25 -2800 28800 -72900 50000
%o A244123 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244123   v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
%o A244123   for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244123   for(k=1, n, v[irow+k]=n*(n-k*b)^(k-1)*(k*b)^(n-k)*binomial(n, k); ); );
%o A244123   return(v); }
%o A244123   a=seq(100,-1);
%Y A244123 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244123 sign,tabl
%O A244123 0,5
%A A244123 _Stanislav Sykora_, Jun 21 2014