A344724 -id:A344724 - cp's OEIS frontend

A344726 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/j)^k.

1, 1, 1, 1, 3, 3, 1, 7, 9, 2, 1, 15, 27, 12, 4, 1, 31, 81, 56, 22, 4, 1, 63, 243, 240, 118, 30, 6, 1, 127, 729, 992, 610, 196, 44, 4, 1, 255, 2187, 4032, 3094, 1230, 324, 48, 7, 1, 511, 6561, 16256, 15562, 7564, 2336, 448, 71, 7, 1, 1023, 19683, 65280, 77998, 45990, 16596, 3840, 685, 83, 9

Offset: 1

Seiichi Manyama, May 27 2021

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  3,   7,   15,   31,    63, ...
  3,  9,  27,   81,  243,   729, ...
  2, 12,  56,  240,  992,  4032, ...
  4, 22, 118,  610, 3094, 15562, ...
  4, 30, 196, 1230, 7564, 45990, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..5 give A059851, A344720, A344721, A344722, A344723.
T(n,n) gives A344724.
Cf. A344725.

T[n_, k_] := Sum[(-1)^(j + 1) * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)

T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\j)^k);

T(n, k) = sum(j=1, n, sumdiv(j, d, (-1)^(j/d+1)*(d^k-(d-1)^k)));

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 + x^j).

T(n,k) = Sum_{j=1..n} Sum_{d|j} (-1)^(j/d + 1) * (d^k - (d - 1)^k).

A350224 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^n).

Original entry on oeis.org

1, 3, 25, 242, 3038, 45981, 817445, 16714015, 386682183, 9990395075, 285173906017, 8913939944658, 302838320218733, 11111330886027346, 437880557043106024, 18446463023081162520, 827233956902648749749, 39346258081658580861101

Offset: 1

Seiichi Manyama, Dec 20 2021

nonn

Cf. A344724, A347416, A350221, A350222, A350223.

a[n_] := Sum[(-1)^(k + 1)*Floor[(n/k)^n], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Dec 20 2021 *)

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A344726 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/j)^k.

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A350224 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^n).

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

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