A097243 -id:A097243 - cp's OEIS frontend

A007096 Expansion of theta_3 / theta_4.

1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832

Offset: 0

N. J. A. Sloane, Simon Plouffe

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005
a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - Jeremy Lovejoy, Aug 29 2020

			G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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, 8, 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, p. 11.
Bernard L.S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electronic J. Combin. 19, 2012, Paper 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005
Cf. A014969, A001936, A001938, A079006, A127391, A127392.
Cf. A022577, A080054, A097243, A123655.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

{a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* Michael Somos, Jul 07 2005*/

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))}; /* Michael Somos, Jan 01 2006 */

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005
Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.
Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005
Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A007248 McKay-Thompson series of class 4C for the Monster group.

Original entry on oeis.org

1, 20, -62, 216, -641, 1636, -3778, 8248, -17277, 34664, -66878, 125312, -229252, 409676, -716420, 1230328, -2079227, 3460416, -5677816, 9198424, -14729608, 23328520, -36567242, 56774712, -87369461, 133321908, -201825396, 303248408, -452431503

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 20*x - 62*x^2 + 216*x^3 - 641*x^4 + 1636*x^5 - 3778*x^6 + ...
T4C = 1/q + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, ``More on replicable functions,'' Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
McKay, John; Strauss, Hubertus. The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra 18 (1990), no. 1, 253-278.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for McKay-Thompson series for Monster simple group

Cf. A029845, A097243, A124972, A112144.

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

a[ n_] := SeriesCoefficient[ 8 q + (QPochhammer[ q] / QPochhammer[ q^4])^8, {q, 0, 2 n}]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ -8 + 16 / m, {q, 0, 2 n - 1}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[ -8 + 16 (EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q])^4, {q, 0, 2 n - 1}]; (* Michael Somos, Jul 22 2011 *)

{a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8*x + (eta(x + A) / eta(x^4 + A))^8, n))}; /* Michael Somos, Nov 14 2006 */

G.f.: 16*(theta_3/theta_2)^4 - 8 = 16 / lambda(z) - 8.
G.f.: 8*x^(1/2) + (Product_{k>0} (1 - x^(k/2)) / (1 - x^(2*k)))^8.
Expansion of q * ( -8 + 16 / lambda(z)) in powers of q^2 where nome q = exp(Pi i z). - Michael Somos, Nov 14 2006
Expansion of 4 * q^(1/2) * (k(q) + 1 / k(q)) in powers of q where nome q = exp(Pi i z). - Michael Somos, Nov 11 2014
Expansion of q * (8 + (eta(q) / eta(q^4))^8) in powers of q^2. - Michael Somos, Nov 14 2006
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v + 24)^2 - (v + 8) * u^2. - Michael Somos, Nov 14 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A097243. - Michael Somos, Jul 22 2011
a(n) = A029845(2*n - 1) = A124972(2*n - 1). - Michael Somos, Nov 14 2006.
a(n) ~ (-1)^(n+1) * exp(sqrt(2*n)*Pi) / (2^(5/4)*n^(3/4)). - Vaclav Kotesovec, Aug 09 2025

A189925 Expansion of theta_4/theta_3 in powers of q.

Original entry on oeis.org

1, -4, 8, -16, 32, -56, 96, -160, 256, -404, 624, -944, 1408, -2072, 3008, -4320, 6144, -8648, 12072, -16720, 22976, -31360, 42528, -57312, 76800, -102364, 135728, -179104, 235264, -307672, 400704, -519808, 671744, -864960, 1109904

Offset: 0

Michael Somos, May 01 2011

sign

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

In Baker [1890] page 94 is equation (1): sqrt(cos theta) = [[...]] = 1 - 4q + 8q^2 -[[...]] where cos theta = k'. - Michael Somos, Dec 31 2023

			G.f. = 1 - 4*q + 8*q^2 - 16*q^3 + 32*q^4 - 56*q^5 + 96*q^6 - 160*q^7 + 256*q^8 + ...

Arthur L. Baker, Elliptic Functions, John Wiley & Sons, NY, 1890.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Berger, Relations between cusp forms on congruence and noncongruence groups, Proc. of the Amer. Math. Soc., vol. 128, (2000), 2869-2874. see p. 2870 equation (4).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A007096, A014969, A079006, A093160.

nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^2 / (1+x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
With[{nmax = 50}, CoefficientList[Series[4 QPochhammer[-1, x^2]^2/QPochhammer[-1, x]^4, {x, 0, nmax}], x]] (* Jan Mangaldan, Jan 04 2017 *)
With[{nmax = 50}, CoefficientList[Series[EllipticTheta[4, 0, x]/EllipticTheta[3, 0, x], {x, 0, nmax}], x]] (* Jan Mangaldan, Jan 04 2017 *)
a[ n_] :=  SeriesCoefficient[(1 - InverseEllipticNomeQ[x])^(1/4), {x, 0, n}]; (* Michael Somos, Dec 31 2023 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^2 * eta(x^4 + A) / eta(x^2 + A)^3 )^2, n))};

Expansion of eta(q)^4 * eta(q^4)^2 / eta(q^2)^6 in powers of q.
Expansion of Jacobian elliptic function sqrt(k') in powers of q.
Expansion of phi(-q) / phi(q) = chi(-q)^2 / chi(q)^2 = psi(-q)^2 / psi(q)^2 = phi(-q)^2 / phi(-q^2)^2 = phi(-q^2)^2 / phi(q)^2 = chi(-q)^4 / chi(-q^2)^2 = chi(-q^2)^2 / chi(q)^4 = f(-q)^2 / f(q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -4, 2, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 * (u^2 + 1) - 2*u.
Unique solution to f(x^2)^(-2) = (f(x) + 1/f(x)) / 2 and f(0) = 1, f'(0) nonzero.
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A079006.
G.f.: theta_4 / theta_3 = (Sum_{k} (-x)^k^2)/(Sum_{k} x^k^2) = (Product_{k>0} ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2 / (1 - x^(4*k - 2)))^2.
Convolution inverse of A007096. a(n) = (-1)^n * A007096(n). a(2*n) = A014969(n). a(2*n + 1) = -4 * A093160(n). a(4*n) = A097243(n). a(4*n + 2) = 8*A022577(n).
a(n) ~ (-1)^n * exp(Pi*sqrt(n))/(2^(5/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: exp(-4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

cp's OEIS Frontend

A007096 Expansion of theta_3 / theta_4.

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A007248 McKay-Thompson series of class 4C for the Monster group.

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A189925 Expansion of theta_4/theta_3 in powers of q.

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