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A244121 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

%I A244121 #17 Jun 25 2014 09:38:35
%S A244121 1,0,1,0,4,0,0,9,18,0,0,16,192,48,0,0,25,1200,1800,100,0,0,36,5760,
%T A244121 29160,11520,180,0,0,49,23520,317520,423360,58800,294,0,0,64,86016,
%U A244121 2721600,9175040,4536000,258048,448,0,0,81,290304,19840464,145152000,181440000,39680928,1016064,648,0
%N A244121 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).
%C A244121 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 A244121 Stanislav Sykora, <a href="/A244121/b244121.txt">Table of n, a(n) for rows 0..100</a>
%H A244121 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 A244121 First rows of the triangle, all summing up to n^n:
%e A244121 1
%e A244121 0 1
%e A244121 0 4   0
%e A244121 0 9  18   0
%e A244121 0 16 192  48  0
%e A244121 0 25 1200 1800 100 0
%o A244121 (PARI) seq(nmax,b)={my(v,n,k,irow);
%o A244121   v = vector((nmax+1)*(nmax+2)/2);v[1]=1;
%o A244121   for(n=1,nmax,irow=1+n*(n+1)/2;v[irow]=0;
%o A244121   for(k=1,n,v[irow+k]=n*(n-k*b)^(k-1)*(k*b)^(n-k)*binomial(n,k);););
%o A244121   return(v);}
%o A244121   a=seq(100,1);
%Y A244121 Cf. A244116, A244117, A244118, A244119, A244120, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244137, A244138, A244139, A244140, A244141, A244142, A244143.
%K A244121 nonn,tabl
%O A244121 0,5
%A A244121 _Stanislav Sykora_, Jun 21 2014