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

1, 0, 1, 0, 3, 4, 0, 7, 32, 27, 0, 15, 176, 405, 256, 0, 31, 832, 3888, 6144, 3125, 0, 63, 3648, 30618, 90112, 109375, 46656, 0, 127, 15360, 216513, 1048576, 2265625, 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

Offset: 0

Author

Olivier Gérard, Aug 02 2012

Keywords

• nonn
• tabl

Comments

Initial term T(0,0) may be computed as 0, depending on formula and convention.

Examples

			      1
      0       1
      0       3       4
      0       7      32      27
      0      15     176     405     256
      0      31     832    3888    6144    3125
      0      63    3648   30618   90112  109375   46656
      0     127   15360  216513 1048576 2265625 2239488  823543
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Row sums sequence is A215077.

Product of A055248 and A089072 (with an initial 0 in each row).

Cf. A000225 (column k=1), A000312 (diagonal).

Programs

Maple

A215079 := proc(n,k)
    k^n*add( binomial(n,n-k-j),j=0..n-k) ;
end proc: # R. J. Mathar, Feb 08 2021

Mathematica

Flatten[Table[Table[Sum[k^n*Binomial[n, n - k - j], {j, 0, n - k}],  {k, 0, n}], {n, 0, 10}], 1]

Formula

T(n,k) = k^n * sum(binomial(n,n-k-j),j=0..n-k) = k^n * A055248(n,k-1).

T(n,k) = k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1)

T(n,1) = A000225(n). - R. J. Mathar, Feb 08 2021