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
KEYWORD
AUTHOR
Olivier Gérard, Aug 02 2012
STATUS
approved