Newton's interpolation formula and sums of powers

Abstract

In this manuscript we derive formulas for multifold sums of powers by utilizing Newton's interpolation formula. Furthermore, we provide formulas for multifold sums of powers in terms of Stirling numbers of the second kind and Eulerian numbers.

Related works

• Newton's interpolation formula and sums of powers (2025)
• Sums of powers via central finite differences and Newton's formula (2025)
• Sums of powers via backward finite differences and Newton's formula (2026)

OEIS

• https://oeis.org/A131689 - Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows, T(n, k) for 0 <= k <= n.
• https://oeis.org/A028246 - Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
• https://oeis.org/A038719 - Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.
• https://oeis.org/A391552 - Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (3+j)^n.
• https://oeis.org/A391633 - Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (4+j)^n.
• https://oeis.org/A391635 - Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (5+j)^n.

Metadata

MSC2010: 05A19, 05A10, 11B83, 03C40.
Keywords: Sums of powers, Newton's interpolation formula, Finite differences, Binomial coefficients, Newton's series, Faulhaber's formula, Bernoulli numbers, Bernoulli polynomials, Interpolation, Combinatorics, Central factorial numbers, OEIS, Stirling numbers, Eulerian numbers, Worpitzky identity.