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

0,5

Comments

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

Links

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

Formula

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

Example

  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

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]

Crossrefs

Binomial convolution of A215083.

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

Sequence in context: A208476 A247449 A112695 * A067881 A307649 A024714

Adjacent sequences: A215075 A215076 A215077 * A215079 A215080 A215081

Keyword

nonn,tabl

Author

Olivier Gérard, Aug 02 2012

Status

approved