A145707 -id: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.

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

A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 584, 735, 923, 1151, 1435, 1775, 2194, 2698, 3311, 4045, 4935, 5994, 7270, 8787, 10600, 12749, 15310, 18330, 21912, 26130, 31107, 36949, 43823, 51863, 61290, 72293, 85145, 100107

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

General asymptotic formula (Hagis, 1971): If s > 1 and g.f. = Product_{k>=1} (1 - x^(s*k))/(1 - x^k), then a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), minor asymptotic terms added by Vaclav Kotesovec, Jan 13 2017
The formula in the article by Noureddine Chair: "The Euler-Riemann Gases, and Partition Identities", p. 32, is incorrect (must be s -> s-1 and 24 -> 24*n).
Number of partitions in which no part occurs more than 9 times. - Ilya Gutkovskiy, May 31 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 2013, p. 32.
Peter Hagis jr., Partitions with a restriction on the multiplicity of the summands, Transactions of the American Mathematical Society, Volume 155, Number 2, April 1971.
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.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261772, A145707, A320612.

nmax = 50; CoefficientList[Series[Product[(1 - x^(10*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 10], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 29 2020 *)

Vec(prod(k=1, 51, (1 - x^(10*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

a(n) ~ 3*Pi * BesselI(1, sqrt((24*n + 9)/10) * Pi/2) / (5*sqrt(24*n + 9)) ~ exp(Pi*sqrt(3*n/5)) * 3^(1/4) / (4 * 5^(3/4) * n^(3/4)) * (1 + (3^(3/2)*Pi/(16*sqrt(5)) - sqrt(15)/(8*Pi)) / sqrt(n) + (27*Pi^2/2560 - 25/(128*Pi^2) - 45/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017

a(n) = (1/n)*Sum_{k=1..n} A284344(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

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)).

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

A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 20, 24, 28, 33, 40, 46, 54, 64, 74, 86, 100, 115, 133, 154, 176, 202, 231, 263, 300, 342, 388, 440, 499, 563, 636, 718, 808, 909, 1022, 1146, 1284, 1439, 1608, 1797, 2006, 2236, 2490, 2772, 3081, 3422, 3800, 4212

Offset: 0

Vaclav Kotesovec, Aug 31 2015

nonn

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 10.

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)) * (1 - (3*sqrt(3*m)/(8*Pi*sqrt(m-1)) + (m-1)^(3/2)*Pi/(48*sqrt(3*m))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

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

Cf. A145707.
Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9).
Column k=10 of A290307.

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 10*j*(j+1))(iquo(i, 10, '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^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(3*n/10)) * 3^(1/4) / (2^(7/4) * 5^(1/4) * n^(3/4)) * (1 - (sqrt(15)/(4*Pi*sqrt(2)) + 3*Pi*sqrt(3)/(16*sqrt(10))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(20*k-10))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017

A145705 Expansion of q^(1/4) * (eta(q^8) * eta(q^10) - eta(q^2) * eta(q^40)) / (eta(q^4) * eta(q^20)) in powers of q.

Original entry on oeis.org

1, -1, 0, 1, 1, 0, 0, 1, 1, -1, -1, 1, 2, -1, -1, 1, 2, -2, -1, 2, 3, -3, -2, 3, 4, -3, -2, 4, 5, -4, -4, 5, 6, -6, -5, 6, 8, -7, -6, 8, 11, -10, -8, 11, 13, -11, -10, 13, 16, -15, -14, 17, 20, -18, -17, 20, 24, -23, -21, 25, 31, -29, -26, 32, 37, -34, -32, 39, 44, -42, -41, 47, 54, -52, -49, 56, 64, -62, -59, 68, 79, -77, -72

Offset: 0

Michael Somos, Oct 17 2008, Nov 11 2008, Jan 21 2009

sign

			1/q - q^3 + q^11 + q^15 + q^27 + q^31 - q^35 - q^39 + q^43 + 2*q^47 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

(-1)^n * A145704(n) = a(n). A145706(n) = a(2*n). - A145707(n) = a(2*n + 1).

QP = QPochhammer; s = (QP[q^8]*QP[q^10]-q*QP[q^2]*QP[q^40])/(QP[q^4]* QP[q^20]) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^8 + A) * eta(x^10 + A) - x * eta(x^2 + A) * eta(x^40 + A)) / (eta(x^4 + A) * eta(x^20 + A)), n))}

Denoted by "(160~a)" in Simon Norton's replicable function list.

G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).

cp's OEIS Frontend

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