A141094 -id:A141094 - cp's OEIS frontend

A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18).

1, -1, 0, 2, -2, -1, 4, -4, -1, 8, -8, -2, 14, -14, -4, 24, -23, -6, 40, -38, -10, 63, -60, -16, 98, -92, -24, 150, -140, -36, 224, -208, -54, 329, -304, -78, 478, -440, -112, 684, -627, -160, 968, -884, -224, 1358, -1236, -312, 1884, -1710, -432, 2592, -2346, -590, 3540, -3196, -801, 4796, -4320, -1082, 6454

Offset: 0

Michael Somos, Mar 07 2004

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - x + 2*x^3 - 2*x^4 - x^5 + 4*x^6 - 4*x^7 - x^8 + 8*x^9 + ...
G.f. = q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A062242, A128111, A141094, A216046.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 - x^k, {k, 3, n, 6}]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)
a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}; /* Michael Somos, Dec 07 2013 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + (x*A^2)^2) - x*A^2); polcoeff(A, n))};

{a(n) = if( n<0, 0, polcoeff( prod(k=0, (n-1)\2, (1 - x^(2*k + 1))^if(k%3==1, -2, 1), 1 + x * O(x^n)), n))};

Expansion of chi(-q) / chi(-q^3)^3 where chi() is a Ramanujan theta function.
Expansion of q^(-1/3) * c(q^2) / c(q) where c() is a cubic AGM theta function. - Michael Somos, Oct 04 2006
Expansion of q^(-1/3) * eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-1, 0, 2, 0, -1, 0, ...].
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^2 - v + 2*u*v^2.
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - v^2 + v) - u^3 * (1 + 2*v + 4*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A141094. - Michael Somos, Dec 07 2013
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3.
G.f.: 1 / (1 + (x + x^2) / (1 + (x^2 + x^4) / (1 + (x^3 + x^6) / ...))).
a(n) = A062242(2*n + 1) = (-1)^n * A128111(n). Convolution inverse of A062242.
a(2*n + 1) = - A216046(n). Convolution square is A216046. - Michael Somos, Dec 07 2013
G.f.: T(0), where T(k) = 1 - (x^(k+1)+x^(2*k+2))/((x^(k+1)+x^(2*k+2))+1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013

A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 6, 12, 21, 36, 60, 96, 150, 228, 342, 504, 732, 1050, 1488, 2088, 2901, 3996, 5460, 7404, 9972, 13344, 17748, 23472, 30876, 40413, 52644, 68268, 88152, 113364, 145224, 185352, 235734, 298800, 377514, 475488, 597108, 747690, 933672, 1162824

Offset: 0

Michael Somos, Feb 15 2007

nonn

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 3*q + 6*q^2 + 12*q^3 + 21*q^4 + 36*q^5 + 60*q^6 + 96*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, 11, 13.
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A062242, A128129, A141094.

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^3 QPochhammer[ q^3] / (QPochhammer[ q]^3 QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
nmax=60; CoefficientList[Series[Product[(1-x^(2*k))^3 * (1-x^(3*k)) / ((1-x^k)^3 * (1-x^(6*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)

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

Expansion of eta(q^2)^3 * eta(q^3) / (eta(q)^3 * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 3, 0, 2, 0, 3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v - 2*u*v^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u + u^2 + u^3) - v^3*(1 - 2*u + 4*u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = u^6 + v^6 - 16*u^5*v^5 + 20*u^4*v^4 + 10*u^2*v^2*(u^3 + v^3) - 20*u^3*v^3 - 5*u*v*(u^3 + v^3) + 5*u^2*v^2 - u*v.
Expansion of b(q^2) / b(q) in powers of q where b() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062242.
a(n) = 3*A128129(n) unless n=0.
Convolution inverse of A141094. - Michael Somos, Feb 19 2015
a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A242405 Expansion of (b(q) / b(q^2))^2 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -6, 15, -24, 39, -72, 123, -192, 294, -456, 693, -1008, 1452, -2100, 2991, -4176, 5781, -7992, 10950, -14808, 19908, -26688, 35541, -46944, 61692, -80826, 105366, -136536, 176208, -226728, 290565, -370704, 471318, -597600, 755217, -950976, 1193988

Offset: 0

Michael Somos, May 13 2014

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 6*q + 15*q^2 - 24*q^3 + 39*q^4 - 72*q^5 + 123*q^6 - 192*q^7 + ...

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

Cf. A141094, A216046.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2]^3 / QPochhammer[ x^3, x^6])^2, {x, 0, n}];

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

Expansion of (chi(-q)^3 / chi(-q^3))^2 in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3)))^2 in powers of q.
Euler transform of period 6 sequence [ -6, 0, -4, 0, -6, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is g.f. for A216046.
G.f.: Product_{k>0} ((1 - x^(2*k-1))^3 / (1 - x^(6*k-3)))^2.
Convolution square of A141094.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)/3) / (2^(3/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A261035 A weighted count of the number of overpartitions of n with restricted odd differences.

Original entry on oeis.org

1, -1, -1, -1, 2, -1, 4, -5, 7, -8, 10, -15, 18, -22, 26, -37, 46, -53, 66, -84, 104, -122, 148, -183, 224, -263, 312, -379, 454, -531, 626, -750, 887, -1034, 1208, -1428, 1672, -1936, 2250, -2633, 3062, -3529, 4076, -4728, 5460, -6264, 7196, -8290, 9520, -10875, 12431, -14238

Offset: 0

Jeremy Lovejoy, Aug 07 2015

sign

The number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part of the overpartition is odd then it is overlined.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

Cf. A260890. Equals the convolution of A141094 and A260984.

G.f.: (Product_{n >= 1} (1+q^(3*n))/(1+q^n)^3) * (1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)*(1+q)^2*(1+q^2)^2*...*(1+q^(n-1))^2*(1+q^n)/((1+q^3)*(1+q^6)*...*(1+q^(3*n)))).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Jun 12 2019

A261037 The number of overpartitions of n with restricted odd differences and smallest part both odd and overlined.

Original entry on oeis.org

1, 1, 3, 4, 7, 10, 17, 23, 36, 48, 73, 96, 140, 182, 259, 334, 463, 592, 806, 1024, 1370, 1728, 2281, 2860, 3727, 4646, 5991, 7430, 9487, 11706, 14822, 18205, 22870, 27966, 34890, 42492, 52670, 63896, 78743, 95178, 116659, 140516, 171380, 205750

Offset: 1

Jeremy Lovejoy, Aug 07 2015

nonn

The number of overpartitions of n such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is both odd and overlined.

K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

Cf. A141094, A260890, A260983, A261035.

G.f.: 1 + 3*Sum_{n >= 1} a(n)*q^n = (Product_{n >= 1} (1-q^(3*n))/((1-q^n)*(1-q^(2*n)))) * (1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)*(1-q^2)*(1-q^4)*...*(1-q^(2*n-2))*(1-q^n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n)))) = A260890(q)*A260983(q).

cp's OEIS Frontend

A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18).

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A128128 Expansion of chi(-q^3) / chi^3(-q) in powers of q where chi() is a Ramanujan theta function.

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A242405 Expansion of (b(q) / b(q^2))^2 in powers of q where b() is a cubic AGM theta function.

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A261035 A weighted count of the number of overpartitions of n with restricted odd differences.

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A261037 The number of overpartitions of n with restricted odd differences and smallest part both odd and overlined.

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