A215083 Triangle T(n,k) = sum of the k first n-th powers

0, 0, 1, 0, 1, 5, 0, 1, 9, 36, 0, 1, 17, 98, 354, 0, 1, 33, 276, 1300, 4425, 0, 1, 65, 794, 4890, 20515, 67171, 0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 0, 1, 257, 6818, 72354, 462979, 2142595, 7907396, 24684612, 0, 1, 513, 20196, 282340, 2235465, 12313161, 52666768, 186884496, 574304985, 0, 1, 1025, 60074, 1108650, 10874275, 71340451, 353815700, 1427557524, 4914341925, 14914341925

Offset

0,6

Comments

First term T(0,0) = 0 can be computed as 1 if one starts the sum at j=0 and take the convention 0^0 = 1.

Links

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

Formula

T(n, k) = Sum_{j=1..k} j^n

Sum_{j=0..n}((-1)^(n-j)/(j+1)*binomial(n+1,j+1)*T(n,j)) are the Bernoulli numbers B(n) = B(n, 1) by a formula of L. Kronecker. - Peter Luschny, Oct 02 2017

Example

Triangle starts (using the convention 0^0 = 1, see the first comment):
[0] 1
[1] 0, 1
[2] 0, 1,  5
[3] 0, 1,  9,  36
[4] 0, 1, 17,  98,  354
[5] 0, 1, 33, 276, 1300,  4425
[6] 0, 1, 65, 794, 4890, 20515, 67171

Maple

A215083 := (n, k) -> add(i^n, i=0..k):
for n from 0 to 8 do seq(A215083(n, k), k=0..n) od; # Peter Luschny, Oct 02 2017

Mathematica

Flatten[Table[Table[Sum[j^n, {j, 1, k}], {k, 0, n}], {n, 0, 10}], 1]
Table[ HarmonicNumber[k, -n], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2013 *)

Crossrefs

Row sums are A215083.

A215078 is the product of this array with the binomial array.

T(3,k) is the beginning of A000537.

T(4,k) is the beginning of A000538.

T(5,k) is the beginning of A000539.

Sequence in context: A127557 A060524 A133843 * A221308 A241855 A221800

Adjacent sequences: A215080 A215081 A215082 * A215084 A215085 A215086

Keyword

nonn,tabl

Author

Olivier Gérard, Aug 02 2012

Status

approved