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A264748 a(n) = Sum_{k = 1..n} (k^n - n^k).

%I A264748 #26 Jul 16 2025 08:04:37
%S A264748 0,-1,-3,14,520,11185,239505,5510652,138456936,3803230815,
%T A264748 113833152565,3695302326650,129479186068128,4874312730972685,
%U A264748 196306448145080385,8425000059348756472,383956514250037779376,18521535576956405481147,942952190208348285876501
%N A264748 a(n) = Sum_{k = 1..n} (k^n - n^k).
%H A264748 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannZetaFunction.html">Riemann Zeta Function</a>
%H A264748 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%F A264748 a(n) =  A031971(n) - A031972(n).
%F A264748 a(n) = ((1 - n)*zeta(-n, n + 1) - n*(n^n - 1) + (n - 1)*zeta(-n))/(n - 1) for n>1, where zeta(s) is the Riemann zeta function and zeta(s, a) is the Hurwitz zeta function.
%F A264748 a(n) ~ n^n / (exp(1) - 1). - _Vaclav Kotesovec_, Jul 16 2025
%e A264748 a(1) = 1^1 - 1^1 = 0;
%e A264748 a(2) = 1^2 - 2^1 + 2^2 - 2^2 = -1;
%e A264748 a(3) = 1^3 - 3^1 + 2^3 - 3^2 + 3^3 - 3^3 = -3;
%e A264748 a(4) = 1^4 - 4^1 + 2^4 - 4^2 + 3^4 - 4^3 + 4^4 - 4^4 = 14;
%e A264748 a(5) = 1^5 - 5^1 + 2^5 - 5^2 + 3^5 - 5^3 + 4^5 - 5^4 + 5^5 - 5^5 = 520, etc.
%t A264748 Table[Sum[k^n - n^k, {k, 1, n}], {n, 1, 20}]
%t A264748 Join[{0}, Table[HarmonicNumber[n, -n] - n (n^n - 1)/(n - 1), {n, 2, 20}]]
%o A264748 (PARI) a(n) = sum(k=1, n, k^n - n^k); \\ _Altug Alkan_, Nov 23 2015
%Y A264748 Cf. A031971, A031972.
%K A264748 sign
%O A264748 1,3
%A A264748 _Ilya Gutkovskiy_, Nov 23 2015