A074209 a(n) = Sum_{i=n+1..2n} i^n.

2, 25, 405, 8418, 216400, 6668779, 240361121, 9936764996, 463893277176, 24148657338925, 1387253043076813, 87185783860333910, 5951020164442347800, 438417132703015536399, 34673851743509883542625

Offset

1,1

Comments

A rapidly growing sequence. An even more rapidly growing sequence, the sum of next n terms of the form i^i, is given in A074309. Sum of first n terms of the form i^n is A031971. Sum of first n terms of the form i^i is A001923.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..351

Formula

From Wesley Ivan Hurt, Jan 28 2021: (Start)

a(n) = Sum_{k=1..n} (n+k)^n.

a(n) = Zeta(-n,n+1) - Zeta(-n,2*n+1), where Zeta is the Hurwitz zeta function. (End)

a(n) ~ (2*n)^n / (1 - exp(-1/2)). - Vaclav Kotesovec, Dec 06 2021

Example

a(2) = 25 = 3^2 + 4^2, a(3) = 405 = 4^3 + 5^3 + 6^3, a(4) = 8418 = 5^4 + 6^4 + 7^4 + 8^4, a(5) = 216400 = 6^4 + 7^5 + 8^5 + 9^5 + 10^5.

Mathematica

Table[Sum[i^n, {i, n+1, 2n}], {n, 20}]

Prog

(PARI) a(n) = sum(k=n+1, 2*n, k^n); \\ Seiichi Manyama, Dec 05 2021

Crossrefs

Cf. A001923, A031971, A074309.

Sequence in context: A367506 A212022 A198710 * A209467 A121252 A322727

Adjacent sequences: A074206 A074207 A074208 * A074210 A074211 A074212

Keyword

nonn

Author

Zak Seidov, Sep 22 2002

Extensions

Name changed by Wesley Ivan Hurt, Jan 28 2021

Status

approved