A293387 -id:A293387 - cp's OEIS frontend

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.

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).

A293389 Expansion of ((eta(q)eta(q^3))/eta(q^2)^2)^2 in powers of q.

Original entry on oeis.org

1, -2, 3, -8, 15, -24, 39, -64, 102, -152, 225, -336, 492, -700, 987, -1392, 1941, -2664, 3630, -4936, 6660, -8896, 11817, -15648, 20604, -26942, 35070, -45512, 58800, -75576, 96777, -123568, 157206, -199200, 251613, -316992, 398148, -498460, 622356, -775216

Offset: 0

Seiichi Manyama, Oct 08 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Main diagonal of A293388.

Cf. A293306, A293387.

nmax = 100; CoefficientList[Series[Product[(1 - x^(3*k))^2 / ((1 + x^k)^4 * (1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2017 *)

G.f.: Product_{k>0} (((1 - x^k)*(1 - x^(3*k)))/(1 - x^(2*k))^2)^2.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 08 2017

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

A293389 Expansion of ((eta(q)eta(q^3))/eta(q^2)^2)^2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

A293389 Expansion of ((eta(q)eta(q^3))/eta(q^2)^2)^2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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.