%I #22 Dec 06 2021 04:17:15

%S 2,25,405,8418,216400,6668779,240361121,9936764996,463893277176,

%T 24148657338925,1387253043076813,87185783860333910,

%U 5951020164442347800,438417132703015536399,34673851743509883542625

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

%C 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.

%H Seiichi Manyama, <a href="/A074209/b074209.txt">Table of n, a(n) for n = 1..351</a>

%F From _Wesley Ivan Hurt_, Jan 28 2021: (Start)

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

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

%F a(n) ~ (2*n)^n / (1 - exp(-1/2)). - _Vaclav Kotesovec_, Dec 06 2021

%e 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.

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

%o (PARI) a(n) = sum(k=n+1, 2*n, k^n); \\ _Seiichi Manyama_, Dec 05 2021

%Y Cf. A001923, A031971, A074309.

%K nonn

%O 1,1

%A _Zak Seidov_, Sep 22 2002

%E Name changed by _Wesley Ivan Hurt_, Jan 28 2021