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

%I A215083 #19 Jul 22 2025 23:21:18
%S A215083 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,
%T A215083 4890,20515,67171,0,1,129,2316,18700,96825,376761,1200304,0,1,257,
%U A215083 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
%N A215083 Triangle T(n,k) = sum of the k first n-th powers.
%C A215083 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.
%H A215083 Vincenzo Librandi, <a href="/A215083/b215083.txt">Table of n, a(n) for n = 0..1000</a>
%F A215083 T(n, k) = Sum_{j=1..k} j^n
%F A215083 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
%e A215083 Triangle starts (using the convention 0^0 = 1, see the first comment):
%e A215083 [0] 1
%e A215083 [1] 0, 1
%e A215083 [2] 0, 1,  5
%e A215083 [3] 0, 1,  9,  36
%e A215083 [4] 0, 1, 17,  98,  354
%e A215083 [5] 0, 1, 33, 276, 1300,  4425
%e A215083 [6] 0, 1, 65, 794, 4890, 20515, 67171
%p A215083 A215083 := (n, k) -> add(i^n, i=0..k):
%p A215083 for n from 0 to 8 do seq(A215083(n, k), k=0..n) od; # _Peter Luschny_, Oct 02 2017
%t A215083 Flatten[Table[Table[Sum[j^n, {j, 1, k}], {k, 0, n}], {n, 0, 10}], 1]
%t A215083 Table[ HarmonicNumber[k, -n], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 05 2013 *)
%Y A215083 Row sums are A215083.
%Y A215083 A215078 is the product of this array with the binomial array.
%Y A215083 T(3,k) is the beginning of A000537.
%Y A215083 T(4,k) is the beginning of A000538.
%Y A215083 T(5,k) is the beginning of A000539.
%Y A215083 Cf. A103438.
%K A215083 nonn,tabl
%O A215083 0,6
%A A215083 _Olivier Gérard_, Aug 02 2012