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A215083
Triangle T(n,k) = sum of the k first n-th powers
10
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
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.
Cf. A103438.
Sequence in context: A127557 A060524 A133843 * A221308 A241855 A221800
KEYWORD
nonn,tabl,changed
AUTHOR
Olivier Gérard, Aug 02 2012
STATUS
approved