A113297 -id:A113297 - cp's OEIS frontend

A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 2, -2, 1, 0, 1, -2, 3, -3, 2, -1, 1, -3, 5, -5, 3, -1, 2, -5, 7, -7, 5, -3, 3, -7, 11, -11, 7, -4, 6, -11, 15, -15, 11, -7, 8, -15, 22, -22, 15, -10, 13, -22, 30, -30, 23, -16, 18, -30, 42, -42, 31, -22, 27, -43, 56, -56, 44, -33, 37, -57, 77, -77, 59, -45, 53, -79, 101, -101, 82, -64

Offset: 0

Michael Somos, Jun 26 2005

sign

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015

			q - q^13 - q^61 + q^73 - q^85 + q^97 - q^133 + 2*q^145 - 2*q^157 + q^169 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

Cf. A098884.

Cf. A081360 (m=2), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

nmax = 100; CoefficientList[Series[Product[(1 + x^(3*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
QP = QPochhammer; s = QP[q]*(QP[q^6]/(QP[q^2]*QP[q^3])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)/eta(x^2+A)/eta(x^3+A), n))}

Euler transform of period 6 sequence [ -1, 0, 0, 0, -1, 0, ...].
G.f.: 1/(Product_{k>0} (1+x^(2k-1)+x^(4k-2))) = Product_{k>0} (1-x^(6k-1))(1-x^(6k-5)) = Product_{k>0} (1-x^k+x^(2k)) (where 1-x+x^2 is 6th cyclotomic polynomial).
Given g.f. A(x), then B(x)=x*A(x^12) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=(v^2+u^4)*(v^2+w^4)-2*v^4*(1-v*u^2*w^2).
Expansion of G(x^6) * H(x) - x * G(x) * H(x^6) where G(), H() are Rogers-Ramanujan functions.
a(n) = (-1)^n*A098884(n).
a(n) ~ (-1)^n * exp(sqrt(n)*Pi/3) / (2*sqrt(6)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
a(n) = -(1/n)*Sum_{k=1..n} A186099(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 26 2017

A145707 Expansion of chi(-q) / chi(-q^10) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 3, -3, 3, -4, 4, -5, 6, -6, 7, -8, 10, -11, 11, -13, 15, -17, 18, -20, 23, -25, 29, -32, 34, -39, 42, -47, 52, -56, 62, -68, 77, -83, 89, -99, 108, -119, 129, -139, 154, -167, 183, -199, 214, -234, 253, -276, 299, -322, 350

Offset: 0

Michael Somos, Oct 17 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 3*x^10 + ...
G.f. = q^3 - q^11 - q^27 + q^35 - q^43 + q^51 - q^59 + 2*q^67 - 2*q^75 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A145703, A145704, A145705.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9).

nmax = 100; CoefficientList[Series[Product[(1 + x^(10*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};

Expansion of q^(-3/8) * eta(q) * eta(q^20) / (eta(q^2) * eta(q^10)) in powers of q.
Euler transform of period 20 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(20*k - 10)).
a(n) = (-1)^n * A145703(n) = A145704(2*n + 1) = - A145705(2*n + 1).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015

A261734 Expansion of Product_{k>=1} (1 + x^(4*k))/(1 + x^k).

Original entry on oeis.org

1, -1, 0, -1, 2, -2, 1, -2, 4, -4, 3, -4, 8, -8, 6, -9, 14, -14, 12, -16, 24, -25, 22, -28, 40, -42, 38, -48, 65, -68, 64, -78, 102, -108, 104, -124, 159, -168, 164, -194, 242, -256, 254, -296, 362, -385, 386, -444, 536, -570, 576, -658, 782, -832, 848, -961

Offset: 0

Vaclav Kotesovec, Aug 30 2015

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

Cf. A081360 (m=2), A109389 (m=3), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

seq(coeff(series(mul((1+x^(4*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, Jul 29 2018

nmax = 100; CoefficientList[Series[Product[(1 + x^(4*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (-1)^n * exp(sqrt(n)*Pi/2) / (4*sqrt(2)*n^(3/4)).

A261736 Expansion of Product_{k>=1} (1 + x^(6*k))/(1 + x^k).

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 2, -2, 2, -3, 3, -3, 5, -5, 5, -7, 8, -8, 11, -12, 12, -16, 17, -18, 23, -25, 26, -32, 35, -37, 45, -49, 52, -62, 67, -72, 85, -92, 98, -114, 124, -133, 153, -166, 178, -203, 220, -236, 268, -290, 311, -350, 379, -407, 456, -493, 529

Offset: 0

Vaclav Kotesovec, Aug 30 2015

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

seq(coeff(series(mul((1+x^(6*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, Jul 29 2018

nmax = 100; CoefficientList[Series[Product[(1 + x^(6*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)).

A133563 Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 1, 0, 0, -1, 1, -1, 2, -2, 2, -2, 2, -1, 2, -3, 2, -3, 5, -5, 4, -5, 6, -4, 4, -7, 7, -7, 10, -11, 10, -12, 12, -10, 12, -15, 14, -16, 22, -22, 20, -24, 26, -22, 24, -30, 31, -33, 40, -43, 42, -46, 48, -45, 50, -58, 58, -63, 77, -79, 76, -86, 92, -86, 92, -107, 110, -116, 134, -141, 142, -154, 160, -157

Offset: 0

Michael Somos, Sep 16 2007

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - Vaclav Kotesovec, Aug 31 2015
Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016

			G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + 2*x^10 - 2*x^11 - 2*x^13 + ...
G.f. = q - q^7 - q^19 + q^25 - q^43 + q^49 - q^55 + 2*q^61 - 2*q^67 + 2*q^73 - ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.

G. C. Greubel, Table of n, a(n) for n = 0..1000
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^5, x^10], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)

{a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A)), n))};

Expansion of q^(-1/6) * eta(q) * eta(q^10) / ( eta(q^2) * eta(q^5) ) in powers of q.
Euler transform of period 10 sequence [ -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (360 t)) = f(t) where q = exp(2 Pi i t).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) + w^2 * (u^2 + v).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(x^q), B(q^9)) where f(u, v, w) = (u^3 + w^3) * (v + v^3) + 2 * v^4 - v^2 + u^3 * w^3 * ( 2 - v^2 ).
Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^5), B(q^10)) where f(u1, u2, u5, u10) = u1^2 * u5^2 + u1^2 * u10^4 + u1 * u2^2 * u5 * u10^2 + u2 * u5^2 * u10^3 + u2^3 * u10^3 - u2^2 * u10^2 - u1^3 * u5^3 - u1^4 * u10^2 - u1^3 * u2^2 * u5 - u1^2 * u2 * u5^2 * u10.
G.f.: Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
G.f.: Product_{k>0} (1 + x^(5*k)) / (1 + x^k).
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/15)) / (2^(5/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015

A261733 Expansion of Product_{k>=1} (1 + x^(9*k))/(1 + x^k).

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, -2, 2, -2, 2, -3, 4, -3, 4, -5, 5, -6, 6, -7, 8, -8, 9, -9, 10, -12, 11, -13, 15, -16, 17, -18, 22, -23, 23, -27, 30, -31, 32, -35, 40, -40, 42, -48, 51, -54, 57, -63, 69, -71, 78, -85, 90, -97, 102, -110, 118, -124, 133

Offset: 0

Vaclav Kotesovec, Aug 30 2015

sign

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A145707 (m=10).

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, -1, 0,
        -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1]
       [1+irem(d, 18)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 01 2015

nmax = 100; CoefficientList[Series[Product[(1 + x^(9*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)/3) / (2 * 3^(3/4) * n^(3/4)).

A261735 Expansion of Product_{k>=1} (1 + x^(8*k))/(1 + x^k).

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -1, 3, -3, 2, -3, 4, -4, 4, -5, 8, -8, 7, -9, 11, -12, 12, -14, 20, -21, 19, -24, 28, -30, 31, -35, 45, -48, 47, -55, 64, -68, 71, -80, 97, -103, 104, -119, 135, -145, 152, -168, 198, -211, 216, -243, 272, -291, 307, -337, 386, -412

Offset: 0

Vaclav Kotesovec, Aug 30 2015

sign

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261733 (m=9), A145707 (m=10).

seq(coeff(series(mul((1+x^(8*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, Jul 29 2018

nmax = 100; CoefficientList[Series[Product[(1 + x^(8*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (-1)^n * exp(sqrt(5*n/6)*Pi/2) * 5^(1/4) / (2^(11/4)*3^(1/4)*n^(3/4)).

A097793 McKay-Thompson series of class 56B for the Monster group.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 10, 12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61, 71, 82, 94, 108, 124, 142, 162, 185, 210, 238, 271, 306, 345, 390, 439, 494, 556, 623, 698, 783, 875, 977, 1092, 1216, 1354, 1508, 1674, 1859, 2064, 2286, 2532, 2803, 3098, 3424

Offset: 0

Michael Somos, Aug 24 2004

nonn

Number of partitions of n into distinct parts not divisible by 7.

Also McKay-Thompson series of class 56C for Monster. - Michel Marcus, Feb 19 2014

			1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 +...
T56B = 1/q + q^3 + q^7 + 2q^11 + 2q^15 + 3q^19 + 4q^23 + 4q^27 +...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
Index entries for McKay-Thompson series for Monster simple group

Cf. A113297.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A261771 (m=8), A112193 (m=9), A261772 (m=10).

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^7]/(QP[q]*QP[q^14])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, 1 + x^k, 1 + A) / prod( k=1, n\7, 1 + x^(7*k), 1 + A), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)), n))}

Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, ...].
Expansion of q^(1/4) * eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)) in powers of q.
G.f.: Product_{k>0} (1 + x^k) / (1 + x^(7*k)).
a(n) ~ exp(Pi*sqrt(2*n/7)) / (2 * 14^(1/4) * n^(3/4)) * (1 - (3*sqrt(7)/ (8*Pi*sqrt(2)) + Pi/(4*sqrt(14))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, -1, 0, -3, 0, 1, -1, 0, -1, 2, -1, 4, 0, 1, -1, 0, -1, 1, -2, 1, -5, 0, 1, -1, 0, -1, 1, 0, 1, -1, 6, 0, 1, -1, 0, -1, 1, -1, 0, -2, 1, -8, 0, 1, -1, 0, -1, 1, -1, 2, -1, 4, 0, 10, 0, 1, -1, 0, -1, 1, -1, 1, -2, 1, -4, 0, -12, 0

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

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

Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
Main diagonal gives A081362.
Cf. A286653, A286656, A290307.

Table[Function[k, SeriesCoefficient[Product[(1 + x^(k i))/(1 + x^i), {i, 1, n}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x^k]/QPochhammer[-1, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

For asymptotics of column k see comment from Vaclav Kotesovec in A145707.

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A109389 Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.

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A145707 Expansion of chi(-q) / chi(-q^10) in powers of q where chi() is a Ramanujan theta function.

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A261734 Expansion of Product_{k>=1} (1 + x^(4*k))/(1 + x^k).

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A261736 Expansion of Product_{k>=1} (1 + x^(6*k))/(1 + x^k).

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A133563 Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

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A261733 Expansion of Product_{k>=1} (1 + x^(9*k))/(1 + x^k).

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A261735 Expansion of Product_{k>=1} (1 + x^(8*k))/(1 + x^k).

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