A290216 -id:A290216 - cp's OEIS frontend

A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji)).

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -2, -1, 0, 1, -1, -2, 3, 1, 0, 1, -1, -2, 0, -1, -1, 0, 1, -1, -2, 0, 5, -5, 1, 0, 1, -1, -2, 0, 1, 1, 9, -1, 0, 1, -1, -2, 0, 1, 9, -12, 3, 2, 0, 1, -1, -2, 0, 1, 4, -4, -3, -20, -2, 0, 1, -1, -2, 0, 1, 4, 6, -15, 31, 16, 2

Offset: 0

Seiichi Manyama, Oct 06 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0,  0, -2, -2, -2, ...
   0, -1,  3,  0,  0, ...
   0,  1, -1,  5,  1, ...
   0, -1, -5,  1,  9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A081362, A293287, A290395.
Rows n=0..1 give A000012, (-1)*A057427.
Main diagonal gives A258210.
Cf. A290216, A293307.

A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji))^2.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 7, 6, 0, 1, 2, 7, 10, 9, 0, 1, 2, 7, 16, 25, 14, 0, 1, 2, 7, 16, 31, 38, 22, 0, 1, 2, 7, 16, 39, 62, 78, 32, 0, 1, 2, 7, 16, 39, 70, 117, 116, 46, 0, 1, 2, 7, 16, 39, 80, 149, 206, 206, 66, 0, 1, 2, 7, 16, 39, 80, 159, 262, 362

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0,  2,  2,  2,  2, ...
   0,  3,  7,  7,  7, ...
   0,  6, 10, 16, 16, ...
   0,  9, 25, 31, 39, ...
   0, 14, 38, 62, 70, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000007, A022567.
Rows n=0 gives A000012.
Main diagonal gives A293378.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: A290217 (m=-1), A290216 (m=1), this sequence (m=2).

A290269 G.f.: Product_{m>0} (1 + x^m + 2x^(2m) + 3x^(3m)).

Original entry on oeis.org

1, 1, 3, 5, 6, 10, 18, 25, 34, 53, 66, 94, 136, 179, 236, 325, 416, 550, 714, 913, 1181, 1530, 1917, 2446, 3091, 3862, 4828, 6051, 7457, 9233, 11465, 14028, 17181, 21099, 25651, 31251, 37948, 45853, 55380, 66833, 80236, 96271, 115496, 137822, 164470, 196094, 232837

Offset: 0

Seiichi Manyama, Oct 06 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Column k=3 of A290216.

m = 30; Vec(prod(k=1, m, 1 + x^k + 2*x^(2*k) + 3*x^(3*k)) + O(x^m)) \\ Michel Marcus, Oct 07 2017

A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(j(2*i-1))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 4, 1, 1, 0, 1, 1, 2, 4, 1, 3, 1, 0, 1, 1, 2, 4, 5, 3, 3, 1, 0, 1, 1, 2, 4, 5, 3, 6, 5, 2, 0, 1, 1, 2, 4, 5, 8, 6, 5, 6, 2, 0, 1, 1, 2, 4, 5, 8, 6, 9, 9, 4, 2, 0, 1, 1, 2, 4, 5, 8, 12, 9, 9, 13

Offset: 0

Seiichi Manyama, Oct 09 2017

			Square array begins:
   1, 1, 1, 1, 1, ...
   0, 1, 1, 1, 1, ...
   0, 0, 2, 2, 2, ...
   0, 1, 1, 4, 4, ...
   0, 1, 1, 1, 5, ...
   0, 1, 3, 3, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A000700, A293304, A293463.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A102186.
Cf. A290216.

max = 12; A[n_, k_] := SeriesCoefficient[Product[(x*(-(k*x^((2*i - 1)*(k + 1) + 1)) - x^((2*i - 1)*(k + 1) + 1) + k*x^((2*i - 1)*(k + 1) + 2*i) + x^(2*i)))/(x^(2*i) - x)^2 + 1, {i, 1, max}], {x, 0, n}]; Flatten[ Table[ A[n - k, k], {n, 0, max}, {k, n, 0, -1}]] (* Jean-François Alcover, Oct 10 2017 *)

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -2, 1, 0, 1, -2, -3, -2, 0, 1, -2, -3, 10, 4, 0, 1, -2, -3, 4, -4, -4, 0, 1, -2, -3, 4, 14, -20, 5, 0, 1, -2, -3, 4, 6, -8, 41, -6, 0, 1, -2, -3, 4, 6, 16, -46, 2, 9, 0, 1, -2, -3, 4, 6, 6, -30, 14, -111, -12, 0, 1, -2, -3, 4, 6, 6, 0, -58

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,   1,  1,  1, ...
   0, -2,  -2, -2, -2, ...
   0,  1,  -3, -3, -3, ...
   0, -2,  10,  4,  4, ...
   0,  4,  -4, 14,  6, ...
   0, -4, -20, -8, 16, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000007, A022597.
Rows n=0 gives A000012.
Main diagonal gives A252650.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: this sequence (m=-2), A290217 (m=-1), A290216 (m=1), A293377 (m=2).

cp's OEIS Frontend

A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji)).

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A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji))^2.

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A290269 G.f.: Product_{m>0} (1 + x^m + 2x^(2m) + 3x^(3m)).

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PARI

A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(j(2*i-1))).

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Mathematica

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

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cp's OEIS Frontend

A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i)).

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A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^2.

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A290269 G.f.: Product_{m>0} (1 + x^m + 2*x^(2*m) + 3*x^(3*m)).

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Author

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Programs

PARI

A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*(2*i-1))).

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Mathematica

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i))^2.

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A290217 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji)).

A293377 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji))^2.

A290269 G.f.: Product_{m>0} (1 + x^m + 2x^(2m) + 3x^(3m)).

A293461 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(j(2*i-1))).

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.