A293307 -id:A293307 - 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.

A293305 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} (-1)^jjx^(j*i)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, -3, 0, 1, 0, 1, -1, 1, -3, 0, -3, 0, 0, 1, -1, 1, -3, 4, 0, 4, 1, 0, 1, -1, 1, -3, 4, 0, 4, -3, 0, 0, 1, -1, 1, -3, 4, -5, 0, -3, 4, 0, 0, 1, -1, 1, -3, 4, -5, 0, -7, -2, -2, 0, 0, 1, -1, 1

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -3, -3, ...
   0,  0,  0,  0,  4, ...
   0,  1, -3,  0,  0, ...

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

Columns k=0..2 give A000007, A010815, A293072.
Rows n=0 gives A000012.
Main diagonal gives A293306.
Cf. A293071, A293307.

nmax = 12;
col[k_] := col[k] = Product[1+Sum[(-1)^j*j*x^(i*j), {j, 1, k}], {i, 1, 2 nmax}] + O[x]^(2 nmax) // CoefficientList[#, x]&;
A[n_, k_] := If[n == 0, 1, If[k == 0, 0, col[k][[n+1]]]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 21 2021 *)

A292577 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} (-1)^jjx^(j*i))^2.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 5, 0, 1, 2, 1, 10, 0, 1, 2, 1, -2, 20, 0, 1, 2, 1, 4, -4, 36, 0, 1, 2, 1, 4, 14, 4, 65, 0, 1, 2, 1, 4, 6, 16, 13, 110, 0, 1, 2, 1, 4, 6, -8, 10, 6, 185, 0, 1, 2, 1, 4, 6, 2, -6, 42, -23, 300, 0, 1, 2, 1, 4, 6, 2, 24, 18, 109, -44, 481, 0, 1

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0,  2,  2,  2,  2, ...
   0,  5,  1,  1,  1, ...
   0, 10, -2,  4,  4, ...
   0, 20,  4, 14,  6, ...
   0, 36, 13, 16, -8, ...

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

Columns k=0..1 give A000007, A000712.
Rows n=0 gives A000012.
Main diagonal gives A293387.
Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^m: this sequence (m=-2), A293307 (m=-1), A293305 (m=1), A293388 (m=2).

A293388 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} (-1)^jjx^(j*i))^2.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -2, -1, 0, 1, -2, 3, 2, 0, 1, -2, 3, -2, 1, 0, 1, -2, 3, -8, 1, 2, 0, 1, -2, 3, -8, 7, -6, -2, 0, 1, -2, 3, -8, 15, -6, 14, 0, 0, 1, -2, 3, -8, 15, -14, 17, -20, -2, 0, 1, -2, 3, -8, 15, -24, 17, -14, 22, -2, 0, 1, -2, 3, -8, 15, -24, 27

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, -2, -8,  -8, ...
   0,  1,  1,  7,  15, ...
   0,  2, -6, -6, -14, ...

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

Columns k=0..1 give A000007, A002107.
Rows n=0 gives A000012.
Main diagonal gives A293389.
Product_{i>0} 1/(1 + Sum_{j=1..k} (-1)^j*j*x^(j*i))^m: A292577 (m=-2), A293307 (m=-1), A293305 (m=1), this sequence (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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A293305 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} (-1)^jjx^(j*i)).

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A292577 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} (-1)^jjx^(j*i))^2.

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A293388 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} (-1)^jjx^(j*i))^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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A293305 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} (-1)^j*j*x^(j*i)).

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Mathematica

A292577 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} (-1)^j*j*x^(j*i))^2.

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A293388 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} (-1)^j*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)).

A293305 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} (-1)^jjx^(j*i)).

A292577 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} (-1)^jjx^(j*i))^2.

A293388 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} (-1)^jjx^(j*i))^2.