A293305 -id:A293305 - cp's OEIS frontend

A290216 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)).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 5, 6, 3, 0, 1, 1, 3, 5, 6, 7, 4, 0, 1, 1, 3, 5, 10, 10, 12, 5, 0, 1, 1, 3, 5, 10, 10, 18, 13, 6, 0, 1, 1, 3, 5, 10, 15, 22, 25, 22, 8, 0, 1, 1, 3, 5, 10, 15, 22, 29, 34, 26, 10, 0, 1, 1, 3, 5

Offset: 0

Seiichi Manyama, Oct 06 2017

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  5,  5, ...
   0, 2, 6,  6, 10, ...
   0, 3, 7, 10, 10, ...

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

Columns k=0..3 give A000007, A000009, A293204, A290269.
Rows n=0 gives A000012.
Main diagonal gives A077285.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: A290217 (m=-1), this sequence (m=1), A293377 (m=2).
Cf. A293305.

A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

Original entry on oeis.org

1, -1, 1, -3, 4, -5, 6, -9, 13, -16, 20, -27, 36, -44, 54, -69, 88, -107, 130, -162, 200, -240, 288, -351, 426, -507, 602, -723, 864, -1019, 1200, -1422, 1681, -1968, 2300, -2700, 3160, -3674, 4266, -4965, 5768, -6665, 7692, -8892, 10260, -11792, 13536, -15552

Offset: 0

Seiichi Manyama, Oct 05 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See p. 15.

Main diagonal of A293305.

Cf. A122792.

nmax = 50; CoefficientList[Series[Product[(1 - x^k) * (1 - x^(3*k)) / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2017 *)

G.f.: Product_{i>0} (1 + Sum_{j>0} (-1)^j*j*q^(j*i)).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Oct 05 2017

A293307 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)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, -1, 5, 0, 1, 1, 0, 2, -1, 7, 0, 1, 1, 0, 2, 5, 3, 11, 0, 1, 1, 0, 2, 1, 3, 3, 15, 0, 1, 1, 0, 2, 1, -5, 0, -1, 22, 0, 1, 1, 0, 2, 1, 0, 0, 11, -8, 30, 0, 1, 1, 0, 2, 1, 0, 10, 7, 25, -8, 42, 0, 1, 1, 0, 2, 1, 0

Offset: 0

Seiichi Manyama, Oct 05 2017

			Square array begins:
   1, 1,  1, 1,  1, ...
   0, 1,  1, 1,  1, ...
   0, 2,  0, 0,  0, ...
   0, 3, -1, 2,  2, ...
   0, 5, -1, 5,  1, ...
   0, 7,  3, 3, -5, ...

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

Columns k=0..2 give A000007, A000041, A293294.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A122792.
Cf. A293293, A293305.

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

A290216 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)).

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A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

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A293307 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)).

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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

A290216 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)).

Views

Author

Keywords

Examples

Links

Crossrefs

A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A293307 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)).

Views

Author

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Examples

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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)^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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Author

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Examples

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A290216 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)).

A293307 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)).

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.