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A244135 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

%I A244135 #8 Jun 25 2014 09:41:40
%S A244135 1,0,1,0,6,-2,0,48,-30,9,0,500,-432,252,-64,0,6480,-6860,5760,-2880,
%T A244135 625,0,100842,-122880,131220,-96000,41250,-7776,0,1835008,-2480058,
%U A244135 3150000,-2981440,1890000,-707616,117649,0,38263752,-56000000,81169704,-92897280,76895000,-42674688,14117880,-2097152
%N A244135 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
%C A244135 T(n,k)=(-k)^(k-1)*(n+k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.
%H A244135 Stanislav Sykora, <a href="/A244135/b244135.txt">Table of n, a(n) for rows 0..100</a>
%H A244135 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.(13), with b=1.
%e A244135 First rows of the triangle, all summing up to n^n:
%e A244135 1,
%e A244135 0, 1,
%e A244135 0, 6, -2,
%e A244135 0, 48, -30, 9,
%e A244135 0, 500, -432, 252, -64,
%e A244135 0, 6480, -6860, 5760, -2880, 625,
%o A244135 (PARI) seq(nmax, b)={my(v, n, k, irow);
%o A244135 v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
%o A244135 for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
%o A244135   for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(n+k*b)^(n-k)*binomial(n,k); ); );
%o A244135 return(v); }
%o A244135 a=seq(100, 1);
%Y A244135 Cf. A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244135 sign,tabl
%O A244135 0,5
%A A244135 _Stanislav Sykora_, Jun 22 2014