A074209 - OEIS

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

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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