OFFSET
0,2
LINKS
Seiichi Manyama, Rows n = 0..139, flattened
José L. Cereceda, Sums of powers of integers and generalized Stirling numbers of the second kind, arXiv:2211.11648 [math.NT], 2022.
Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222 [math.CA], 1992.
Petro Kolosov, Formulas for Sums of Powers, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.
Petro Kolosov, Newton's interpolation formula and sums of powers, Zenodo, 2025.
FORMULA
Let F(t, n, k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k, j) * (t + j)^n, then
F(0, n, k) = A131689(n, k),
F(1, n, k) = A028246(n-1, k-1),
F(2, n, k) = A038719(n, k),
F(3, n, k) = T(n, k) (this sequence),
F(4, n, k) = A391633(n, k),
F(5, n, k) = A391635(n, k).
T(n, k) = Sum_{j=0..n} binomial(3, j)*Stirling2(n, k+j)*(k + j)!.
Let S_m(n) be the sum of powers, S_m(n) = 1^m + 2^m + 3^m + ... + n^m, then
S_m(n) = Sum_{j=0..m} T(m, j)*(binomial(n-2, j+1) + (-1)^j * binomial(j+2, j+1) ).
EXAMPLE
Triangle begins:
k= 0 1 2 3 4 5 6 7
-------------------------------------------------------------
n=0: 1;
n=1: 3, 1;
n=2: 9, 7, 2;
n=3: 27, 37, 24, 6;
n=4: 81, 175, 194, 108, 24;
n=5: 243, 781, 1320, 1230, 600, 120;
n=6: 729, 3367, 8162, 11340, 9120, 3960, 720;
n=7: 2187, 14197, 47544, 92526, 109200, 77280, 30240, 5040;
...
MATHEMATICA
(* Prints the values of T(n, k) as triangle. *)
T[t_, n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*(t + j)^n, {j, 0, n}];
Column[Table[T[3, n, k], {n, 0, 10}, {k, 0, n}]]
(* The formula in terms of Stirling2 numbers. *)
T2[t_, n_, k_] := Sum[Binomial[t, j]* StirlingS2[n, k + j]*(k + j)!, {j, 0, n}];
Column[Table[T2[3, n, k], {n, 0, 10}, {k, 0, n}]]
CROSSREFS
KEYWORD
AUTHOR
Petro Kolosov, Dec 13 2025
STATUS
approved