A261733 -id:A261733 - 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

A104502 Number of partitions where no part is a multiple of 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 354, 447, 569, 712, 896, 1113, 1388, 1712, 2117, 2595, 3186, 3882, 4735, 5739, 6959, 8392, 10121, 12150, 14582, 17429, 20823, 24789, 29494, 34979, 41456, 48993, 57856, 68148, 80204

Offset: 0

Eric W. Weisstein, Mar 11 2005

nonn

Coefficients of the B-Dyson Mod 27 identity.

Also partitions where parts are repeated at most 8 times. - Joerg Arndt, Dec 31 2012

			G.f. = 1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...
B(q) = q + q^4 + 2*q^7 + 3*q^10 + 5*q^13 + 7*q^16 + 11*q^19 + 15*q^22 + ...

F. J. Dyson, A walk through Ramanujan's garden, pp. 7-28 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988, see p. 15, eq. (11).

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. 15.
Eric Weisstein's World of Mathematics, Dyson Mod 27 Identities

Cf. A062246, A104501, A104503, A104504.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
Cf. A112193, A261733, A320611.

seq(coeff(series(mul((1-x^(9*k))/(1-x^k),k=1..n),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Sep 29 2018

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
a[n_] := a[n] = (1/n) Sum[DivisorSum[k, Boole[!Divisible[#, 9]] #&] a[n-k], {k, 1, n}]; a[0] = 1;
a /@ Range[0, 50] (* Jean-François Alcover, Oct 01 2019, after Seiichi Manyama *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 9], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 29 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^9 + A) / eta(x + A), n))}; /* Michael Somos, Jan 09 2006 */

{A116607(n)=sigma(n)-if(n%9==0, 9*sigma(n/9))}
{a(n)=polcoeff(exp(sum(k=1, n+1, A116607(k)*x^k/k+x*O(x^n))), n)} /* Paul D. Hanna, Jun 13 2011 */

Expansion of q^(-1/3) * eta(q^9) / eta(q) in powers of q. - Michael Somos, Jan 09 2006
Euler transform of period 9 sequence [1, 1, 1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 09 2006
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^3 + v^3 - u*v - 3*(u*v)^2. - Michael Somos, Jan 09 2006
G.f.: Product_{k>0} (1-x^(9k))/(1-x^k) = 1 + 1/(1-x)*(Sum_{k>0} x^(k^2+k) Product_{i=1..k} (1+x^i+x^(2i))/((1-x^(2i))*(1-x^(2i+1))))
G.f. A(x) = 1/g.f. A062246.
Logarithmic derivative yields A116607 (sum of the divisors of n which are not divisible by 9). - Paul D. Hanna, Jun 13 2011
a(n) ~ 2*Pi * BesselI(1, 4*sqrt(3*n + 1) * Pi/9) / (9*sqrt(3*n + 1)) ~ exp(4*Pi*sqrt(n/3)/3) / (sqrt(2) * 3^(7/4) * n^(3/4)) * (1 + (2*Pi/(9*sqrt(3)) - 9*sqrt(3)/(32*Pi)) / sqrt(n) + (2*Pi^2/243 - 405/(2048*Pi^2) - 5/16) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series that satisfies f(-1 / (81 t)) = 1/3 g(t) where g() is the g.f. for A062246. - Michael Somos, Jun 27 2017

Simplified definition. - N. J. A. Sloane, Oct 20 2019

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

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)).

A113297 Expansion of chi(-q) / chi(-q^7) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, 0, 1, -2, 1, -1, 2, -2, 3, -3, 3, -4, 4, -4, 5, -4, 4, -6, 6, -7, 7, -8, 11, -11, 10, -12, 14, -15, 15, -14, 17, -20, 19, -21, 24, -26, 30, -31, 32, -37, 38, -40, 45, -44, 47, -54, 56, -60, 64, -68, 79, -83, 83, -92, 100, -105, 110, -112, 123, -136, 138, -147, 160, -170, 185, -194, 203

Offset: 0

Michael Somos, Oct 23 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
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^8 - 2*x^9 + x^10 - x^11 + ...
G.f. = q - q^5 - q^13 + q^17 - q^21 + q^25 + q^33 - 2*q^37 + q^41 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
F. G. Garvan and H. Yesilyurt, Shifted and shiftless partition identities II, arXiv:math/0605317 [math.NT], 2003.
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. A097793.

Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A261735 (m=8), A261733 (m=9), A145707 (m=10).

seq(coeff(series(mul((1+x^(7*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..80); # Muniru A Asiru, Jul 29 2018

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^14] / (QPochhammer[ x^2] QPochhammer[ x^7]), {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^14 + A) / (eta(x^2 + A) * eta(x^7 + A)), n))};

Expansion of q^(-1/4) * eta(q) * eta(q^14) / ( eta(q^2) * eta(q^7) ) in powers of q.
Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, ...].
G.f. A(x) = G(x^7) * H(x^2) - x * G(x^2) * H(x^7) where G(x) and H(x) are the Rogers-Ramanujan functions.
G.f.: Product_{k>0} (1 + x^(7*k)) / (1 + x^k).
Expansion of chi(-q) / chi(-q^7) in powers of q where chi() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (224 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} P14(x^k) where P14 is the 14th cyclotomic polynomial.
Convolution inverse is A097793.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/7)) / (2^(3/2) * 7^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015

A112193 Coefficients of replicable function number "54b".

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 95, 110, 127, 145, 167, 191, 218, 249, 283, 322, 365, 414, 469, 529, 597, 673, 757, 851, 955, 1071, 1199, 1341, 1499, 1673, 1865, 2078, 2313, 2572, 2857, 3171, 3517, 3897

Offset: 0

Michael Somos, Aug 28 2005

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 9. - Joerg Arndt, Aug 31 2015

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

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + ... _Michael Somos_, Oct 06 2019
G.f. = q^-1 + q^2 + q^5 + 2*q^8 + 2*q^11 + 3*q^14 + 4*q^17 + 5*q^20 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Index entries for McKay-Thompson series for Monster simple group

Cf. A261733.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A261772 (m=10).

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 9*j*(j+1))(iquo(i, 9, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(9*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
b[n_, i_] := b[n, i] = Module[{q, r}, {q, r} = QuotientRemainder[i, 9]; If[2*n > i*(i+1) - 9*q*(q+1), 0, If[n == 0, 1, b[n, i-1] + If[i>n || r == 0, 0, b[n-i, i-1]]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 07 2016, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^9] / (QPochhammer[ q] QPochhammer[ q^18]), {q, 0, n}]; (* Michael Somos, Oct 06 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) / (eta(x + A) * eta(x^18 + A)), n))}; /* Michael Somos, Oct 06 2019 */

a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)) * (1 - (9*sqrt(3)/ (16*Pi*sqrt(2)) + sqrt(2)*Pi/(9*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017
From Michael Somos, Oct 06 2019: (Start)
Expansion of q^(1/3) * eta(q^2) * eta(q^9) / (eta(q) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...].
G.f. is a period 1 Fourier Series which satisifies f(-1 / (18 t)) = f(t) where q = exp(2 Pi i t).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + u*v) * (u^3 + v^3) - u*v * (1 + u^2*v^2).
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (w^2 - v) * (u^2 - v) - 2*u*v*w.
Convolution inverse of A261733.
(End)

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.

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.

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A104502 Number of partitions where no part is a multiple of 9.

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

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