A350167 -id:A350167 - cp's OEIS frontend

A350164 a(n) = Sum_{k=1..n}(-1)^(k+1) * floor(n/(2*k-1))^n.

1, 4, 26, 255, 3125, 46593, 823415, 16776960, 387400807, 9999941975, 285311495511, 8916083675135, 302875039491581, 11112006557122561, 437893859877597389, 18446743921164642176, 827240261123526320144, 39346407973736968327497

Offset: 1

Seiichi Manyama, Dec 18 2021

nonn

Main diagonal of A350161.

Cf. A101455, A344724, A350145, A350167.

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

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

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

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^n - (d - 1)^n).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 + x^(2*k)).
a(n) ~ n^n. - Vaclav Kotesovec, Dec 18 2021

A350238 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/(2*k-1))^k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 3, 4, 4, 4, 3, 4, 6, 4, 3, 4, 6, 4, 3, 6, 5, 5, 6, 5, 7, 4, 4, 4, 5, 3, 6, 5, 5, 6, 5, 5, 4, 6, 5, 5, 8, 7, 6, 4, 7, 6, 3, 5, 6, 6, 5, 5, 8, 5, 5, 4, 4, 6, 5, 3, 6, 4, 7, 6, 8, 6, 5, 5, 7, 6, 4, 4, 7, 4, 5, 9, 8, 5, 6, 6, 8, 7, 3, 6, 6, 7, 7, 4, 9, 10, 3, 8, 4, 7, 6, 8, 10, 6, 4, 11, 7, 7, 5, 7, 11, 6, 8, 8, 9

Offset: 1

Seiichi Manyama, Dec 21 2021

sign

a(895) = -5.

			a(3) = [3/1] - [(3/3)^2] = 3 - 1 = 2.
a(4) = [4/1] - [(4/3)^2] = 4 - 1 = 3.
a(5) = [5/1] - [(5/3)^2] + [(5/5)^3] = 5 - 2 + 1 = 4.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A350147, A350167, A350223, A350239.

a[n_] := Sum[(-1)^(k+1) * Floor[(n/(2*k-1))^k], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Dec 21 2021 *)

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

cp's OEIS Frontend

A350164 a(n) = Sum_{k=1..n}(-1)^(k+1) * floor(n/(2*k-1))^n.

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A350238 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/(2*k-1))^k).

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