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A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).

%I A215079 #13 Jun 02 2025 08:06:12
%S A215079 1,0,1,0,3,4,0,7,32,27,0,15,176,405,256,0,31,832,3888,6144,3125,0,63,
%T A215079 3648,30618,90112,109375,46656,0,127,15360,216513,1048576,2265625,
%U A215079 2239488,823543,0,255,63232,1436859,10682368,36328125,62145792,51883209,16777216,0,511,257024,9172278,100139008,500000000,1310100480,1856265922,1342177280,387420489,0,1023,1037312,57159432,889192448,6230468750,23339943936,49715643824,60129542144,38354628411,10000000000
%N A215079 Triangle T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k).
%C A215079 Initial term T(0,0) may be computed as 0, depending on formula and convention.
%H A215079 Vincenzo Librandi, <a href="/A215079/b215079.txt">Table of n, a(n) for n = 0..1000</a>
%F A215079 T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k) = k^n * A055248(n,k-1).
%F A215079 T(n,k) = k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1)
%F A215079 T(n,1) = A000225(n). - _R. J. Mathar_, Feb 08 2021
%e A215079       1
%e A215079       0       1
%e A215079       0       3       4
%e A215079       0       7      32      27
%e A215079       0      15     176     405     256
%e A215079       0      31     832    3888    6144    3125
%e A215079       0      63    3648   30618   90112  109375   46656
%e A215079       0     127   15360  216513 1048576 2265625 2239488  823543
%p A215079 A215079 := proc(n,k)
%p A215079     k^n*add( binomial(n,n-k-j),j=0..n-k) ;
%p A215079 end proc: # _R. J. Mathar_, Feb 08 2021
%t A215079 Flatten[Table[Table[Sum[k^n*Binomial[n, n - k - j], {j, 0, n - k}],  {k, 0, n}], {n, 0, 10}], 1]
%Y A215079 Row sums sequence is A215077.
%Y A215079 Product of A055248 and A089072 (with an initial 0 in each row).
%Y A215079 Cf. A000225 (column k=1), A000312 (diagonal).
%K A215079 nonn,tabl
%O A215079 0,5
%A A215079 _Olivier Gérard_, Aug 02 2012