A294758 -id:A294758 - cp's OEIS frontend

A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j) in powers of x.

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -23, 0, 1, -1, -64, -713, -229, 1, 1, -1, -256, -19619, -64807, -2761, 0, 1, -1, -1024, -531185, -16757533, -9688425, -42615, 1, 1, -1, -4096, -14347883, -4294435855, -30499541197, -2165979799, -758499, 0

Offset: 0

Seiichi Manyama, Nov 06 2017

			Square array begins:
    1,    1,      1,         1,           1, ...
   -1,   -1,     -1,        -1,          -1, ...
   -1,   -4,    -16,       -64,        -256, ...
    0,  -23,   -713,    -19619,     -531185, ...
    0, -229, -64807, -16757533, -4294435855, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..1 give A010815, A292312.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000302.
Cf. A283675, A294758.

rows = 10;
col[k_] := col[k] = CoefficientList[Product[(1 - j^(k*j)*x^j), {j, 1, rows + 3}] + O[x]^(rows + 3), x];
A[n_, k_] := col[k][[n + 1]];
(* or: *)
A[0, ] = 1; A[n, k_] := A[n, k] = -(1/n)*Sum[DivisorSum[j, #^(1 + k*j) &]*A[n - j, k], {j, 1, n}];
Table[A[n - k, k], {n, 0, rows - 1}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2017 *)

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j)) * A(n-j,k) for n > 0.

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 9, 6, 1, 1, 33, 90, 13, 1, 1, 129, 2220, 1162, 24, 1, 1, 513, 59178, 265132, 17435, 48, 1, 1, 2049, 1594836, 67180330, 49163241, 310193, 86, 1, 1, 8193, 43048770, 17181660628, 152662629227, 13121450895, 6286826, 160

Offset: 0

Seiichi Manyama, Nov 12 2017

			Square array begins:
    1,     1,        1,            1,               1, ...
    1,     1,        1,            1,               1, ...
    3,     9,       33,          129,             513, ...
    6,    90,     2220,        59178,         1594836, ...
   13,  1162,   265132,     67180330,     17181660628, ...
   24, 17435, 49163241, 152662629227, 476855156157129, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A000219, A294813, A294954.
Rows n=0+1, 2 give A000012, A087289.
Cf. A294609, A294758, A294808.

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

cp's OEIS Frontend

A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j) in powers of x.

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A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j in powers of x.

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cp's OEIS Frontend

A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j) in powers of x.

Views

Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

Formula

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(k*j)*x^j)^j in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294653 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j) in powers of x.

A294950 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-j^(kj)x^j)^j in powers of x.