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a(n) = Sum_{k=1..n}(-1)^(k+1) * floor(n/(2*k-1))^n.
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%I #19 Dec 18 2021 11:17:38

%S 1,4,26,255,3125,46593,823415,16776960,387400807,9999941975,

%T 285311495511,8916083675135,302875039491581,11112006557122561,

%U 437893859877597389,18446743921164642176,827240261123526320144,39346407973736968327497

%N a(n) = Sum_{k=1..n}(-1)^(k+1) * floor(n/(2*k-1))^n.

%F a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^n - (d - 1)^n).

%F a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 + x^(2*k)).

%F a(n) ~ n^n. - _Vaclav Kotesovec_, Dec 18 2021

%t a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^n, {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Dec 18 2021 *)

%o (PARI) a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^n);

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(d^n-(d-1)^n)));

%Y Main diagonal of A350161.

%Y Cf. A101455, A344724, A350145, A350167.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 18 2021