A344721 -id:A344721 - 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).

A350222 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^3).

Original entry on oeis.org

1, 7, 25, 57, 114, 194, 310, 460, 658, 902, 1200, 1557, 1981, 2474, 3045, 3690, 4431, 5257, 6186, 7212, 8350, 9601, 10969, 12461, 14088, 15844, 17746, 19791, 21990, 24343, 26859, 29539, 32399, 35434, 38655, 42060, 45671, 49468, 53480, 57698, 62137, 66793, 71681

Offset: 1

Seiichi Manyama, Dec 20 2021

nonn

Cf. A000578, A309082, A344675, A344721, A350221, A350224.

a[n_] := Sum[(-1)^(k + 1)*Floor[(n/k)^3], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 20 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n^3\k^3));

def A350222(n): return (m:=n**3)+sum(m//k**3 if k&1 else -(m//k**3) for k in range(2,n+1)) # Chai Wah Wu, Oct 27 2023

a(n) = A309082(n^3).

A350163 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^3.

Original entry on oeis.org

1, 8, 26, 63, 125, 209, 335, 504, 703, 981, 1311, 1671, 2141, 2681, 3269, 3990, 4808, 5643, 6669, 7847, 8963, 10343, 11861, 13349, 15212, 17170, 19078, 21310, 23748, 26172, 28962, 31939, 34759, 38133, 41769, 45190, 49188, 53400, 57396, 62246, 67168, 71704, 77122

Offset: 1

Seiichi Manyama, Dec 18 2021

nonn

Column 3 of A350161.

Cf. A101455, A344721, A350144.

a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 18 2021 *)

a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^3);

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(d^3-(d-1)^3)));

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^(2*k)))/(1-x))

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^3 - (d - 1)^3).

G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k-1)^3) * x^k/(1 + x^(2*k)).

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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A350222 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor((n/k)^3).

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A350163 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^3.

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