A215078 -id:A215078 - cp's OEIS frontend

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

Olivier Gérard, Aug 02 2012

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.

			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

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

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.

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

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 *)

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

A215077 Binomial convolution of sum of consecutive powers.

Original entry on oeis.org

0, 1, 7, 66, 852, 14020, 280472, 6609232, 179317056, 5505532992, 188717617280, 7143999854464, 296013377405440, 13325516967972352, 647610246703508480, 33794224057227356160, 1884620857353101983744, 111857608180484932648960, 7040178644779119413723136, 468349192560992552808841216, 32836927387372039917034405888

Offset: 0

Olivier Gérard, Aug 02 2012

a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.

This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs).

Winston de Greef, Table of n, a(n) for n = 0..372 (first 201 terms from Vincenzo Librandi)

Row sums of A215078, A215079, A215080.

See also A215083 and A215084.

Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=sum(k=0,n, binomial(n,k)*sum(j=1,k, j^n)) \\ Charles R Greathouse IV, Jul 31 2016

a(n)=my(P=sumformal('x^n)); sum(k=0,n, binomial(n,k)*subst(P,'x,k)) \\ Charles R Greathouse IV, Jul 31 2016

a(n) = Sum_{k=0..n} binomial(n,k)*Sum_{j=1..k} j^n;
a(n) = Sum_{k=0..n} binomial(n,k)*H_k^{-n}, where H_k^(-n) = k-th harmonic number of order -n;
a(n) = Sum_{k=0..n} k^n * Sum_{j=0..n-k} binomial(n,n-k-j);
a(n) = Sum_{k=0..n} k^n * binomial(n,n-k) * 2F1(1, k-n; k+1)(-1);
a(n) = Sum_{k=0..n} Sum_{j=0..k} (k-j)^n * binomial(n,j);
a(n) = Sum_{k=0..n} Sum_{j=0..n} (n-j)^n * binomial(n,n+k-j);
and the equivalent formulas obtained by symmetries of the binomial and the hypergeometric function as well as treating the zeroth term separately.
a(n) ~ n^n / (sqrt(1+r) * (1-r) * exp(n) * r^n), where r = A202357 = LambertW(exp(-1)). - Vaclav Kotesovec, Jun 10 2019

cp's OEIS Frontend

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

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