A293306 -id:A293306 - cp's OEIS frontend

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)).

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 *)

A122792 Expansion of eta(q^2)^2/(eta(q)eta(q^3)) in powers of q.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 2, 0, 6, 4, 0, 10, 6, 0, 16, 9, 0, 24, 14, 0, 36, 20, 0, 52, 29, 0, 74, 42, 0, 104, 58, 0, 144, 80, 0, 198, 110, 0, 268, 148, 0, 360, 198, 0, 480, 264, 0, 634, 347, 0, 832, 454, 0, 1084, 592, 0, 1404, 764, 0, 1808, 982, 0, 2316, 1257, 0, 2952, 1598, 0

Offset: 0

Michael Somos, Sep 11 2006

nonn

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

A098151(n)=a(3n). A097197(n)=a(3n+1).

Cf. A293306.

QP = QPochhammer; s = QP[q^2]^2/(QP[q]*QP[q^3]) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2/eta(x+A)/eta(x^3+A), n))}

Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, 0, ...].
G.f.: Product_{k>0} (1-x^k)^2/(1+x^k+x^(2k)). a(3n+2)=0.
G.f.: Product_{i>0} 1/(1 + Sum_{j>0} (-1)^j*j*q^(j*i)). - Seiichi Manyama, Oct 08 2017

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

A385520 Expansion of Product_{k>0} ((1 - q^(2k))(1 - q^(6k))^3)/((1 - q^k)(1 - q^(3k))(1 - q^(4k))(1 - q^(12*k))).

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, 17844, 20407

Offset: 0

James Sellers, Jul 01 2025

nonn

a(n) is the number of partitions of the integer n wherein each even part can appear at most twice, while each odd part can appear once, thrice, or four times.

Also, a(n) is the unsigned version of the sequence given in A293306 (this can be seen by replacing q by -q in the generating function).

			For n = 4, the a(4) = 4 partitions are 4, 3+1, 2+2, and 1+1+1+1.  Note that there is one other partition of 4 which is NOT counted by a(4); that is the partition 2+1+1. This partition is NOT counted by a(4) because the odd part 1 appears twice, and this is not allowed from the description given above.

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 pp. 4, 15.

Cf. A293306.

p:=product((1-q^(2*k))*(1-q^(6*k))^3/((1-q^k)*(1-q^(3*k))*(1-q^(4*k))*(1-q^(12*k))), k=1..1000): s:=series(p,q,1000):

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k)) * (1 - x^(6*k))^3 / ((1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2025 *)

my(N=50,q='q+O('q^N)); Vec((eta(-q)*eta(-q^3))/eta(q^2)^2) \\ Joerg Arndt, Jul 02 2025

G.f.: Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).
G.f.: (eta(-q)*eta(-q^3))/eta(q^2)^2.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Jul 02 2025

cp's OEIS Frontend

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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A122792 Expansion of eta(q^2)^2/(eta(q)eta(q^3)) in powers of q.

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

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A385520 Expansion of Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).

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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)).

A385520 Expansion of Product_{k>0} ((1 - q^(2k))(1 - q^(6k))^3)/((1 - q^k)(1 - q^(3k))(1 - q^(4k))(1 - q^(12*k))).