A215078 Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.

0, 0, 1, 0, 2, 5, 0, 3, 27, 36, 0, 4, 102, 392, 354, 0, 5, 330, 2760, 6500, 4425, 0, 6, 975, 15880, 73350, 123090, 67171, 0, 7, 2709, 81060, 654500, 2033325, 2637327, 1200304, 0, 8, 7196, 381808, 5064780, 25926824, 59992660, 63259168, 24684612, 0, 9, 18468, 1696464, 35574840, 281668590, 1034305524, 1896003648, 1681960464, 574304985, 0, 10, 46125, 7208880, 232816500, 2740317300, 14981494710, 42457884000, 64240088580, 49143419250, 14914341925

Offset: 0

Author

Olivier Gérard, Aug 02 2012

Keywords

• nonn
• tabl

Comments

If one starts the sum at j=0, the initial term T(0,0) is 1.

Examples

			  0
  0  1
  0  2    5
  0  3   27       36
  0  4  102      392      354
  0  5  330     2760     6500     4425
  0  6  975    15880    73350   123090    67171
  0  7 2709    81060   654500  2033325  2637327  1200304
		

Links

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

Crossrefs

Binomial convolution of A215083.

Cf. A215077 (row sums), A031971 (diagonal)

Programs

Maple

A215078 := proc(n,k)
        binomial(n,k)*add(j^n,j=1..k) ;
end proc:
seq(seq(A215078(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jan 27 2023

Mathematica

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

Formula

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