A007096 -id:A007096 - cp's OEIS frontend

A080054 G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).

1, 2, 2, 4, 6, 8, 12, 16, 22, 30, 40, 52, 68, 88, 112, 144, 182, 228, 286, 356, 440, 544, 668, 816, 996, 1210, 1464, 1768, 2128, 2552, 3056, 3648, 4342, 5160, 6116, 7232, 8538, 10056, 11820, 13872, 16248, 18996, 22176, 25844, 30068, 34936, 40528

Offset: 0

Michael Somos, Jan 26 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. for pairs of partitions of type R.
G.f. for the number of partitions of 2n in which all odd parts occur with multiplicity 2 and the even parts occur with multiplicity 1. Also g.f. for the number of partitions of 2n free of multiples of 4. All odd parts occur with even multiplicities. The even parts occur with multiplicity 1. - Noureddine Chair, Feb 10 2005
This is also the number of overpartitions of an integer into odd parts. - James Sellers, Feb 18 2008
The Higher Algebra reference on page 517 has an unnumbered example between 251 and 252: "If u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0, prove that (u^2-v^2)^6=16u^2v^2(1-u^8)(1-v^8). [PEMB. COLL. CAMB.]". It turns out that this is two forms of the modular equation of degree 5. - Michael Somos, May 12 2011
Convolution of A000009 and A000700. - Vaclav Kotesovec, Aug 23 2015
Let F(x) = Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)). Let m be a nonzero integer. The simple continued fractions expansions of the real numbers F(1/m) may be predictable - given a positive integer n, the sequence of the n-th partial denominators of the continued fraction expansion of F(1/m) may be polynomial or quasi-polynomial in m for sufficiently large m. An example is given below. - Peter Bala, Nov 03 2019

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 6*q^4 + 8*q^5 + 12*q^6 + 16*q^7 + 22*q^8 + 30*q^9 + ...
From _Peter Bala_, Nov 03 2019: (Start)
F(x) := Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)).
Simple continued fraction expansions of F(1/(2*m)):
  m=2 [1; 1, 2, 1, 1, 1, 1, 2, 1, 2,   33, 1, 3,  7, 4,  33, 1, 8,  4,    2, 1,...]
  m=3 [1; 2, 2, 2, 1, 1, 2, 2, 2, 2,  110, 1, 2, 46, 3, 110, 1, 3, 12,    1, 7,...]
  m=4 [1; 3, 2, 3, 1, 1, 3, 2, 3, 2,  259, 1, 1,  1, 2,  15, 2, 1,  2,  259, 1,...]
  m=5 [1; 4, 2, 4, 1, 1, 4, 2, 4, 2,  504, 1, 1,  1, 1,  78, 1, 1,  2,  504, 1,...]
  m=6 [1; 5, 2, 5, 1, 1, 5, 2, 5, 2,  869, 1, 1,  2, 2,  23, 2, 2,  2,  869, 1,...]
  m=7 [1; 6, 2, 6, 1, 1, 6, 2, 6, 2, 1378, 1, 1,  2, 1, 110, 1, 2,  2, 1378, 1,...]
  m=8 [1; 7, 2, 7, 1, 1, 7, 2, 7, 2, 2055, 1, 1,  3, 2,  31, 2, 3,  2, 2055, 1,...]
  m=9 [1; 8, 2, 8, 1, 1, 8, 2, 8, 2, 2924, 1, 1,  3, 1, 142, 1, 3,  2, 2924, 1,...]
The sequence of the 10th partial denominators [33,110,259,504,...], starting at m = 2, appears to be given by the polynomial 4*m^3 + m - 1.
The sequence of the 15th partial denominators [15,78,23,110,31,142,...], starting at m = 4, appears to be quasi-polynomial in m, with constituent polynomials 4*m - 1 and 16*m - 2. (End)

B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed., G. Bell and Sons, 1895, p. 245, Art. 333.
J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. VI
J. W. L. Glaisher, On Some Continued Fractions, Messenger of Mathematics, 7 (1878), pp. 67-68, see p. 68
H. S. Hall and S. R. Knight, Higher Algebra, Macmillan, 1957, p. 517.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eddy Ardonne, Rinat Kedem, and Michael Stone, Filling the Bose sea: symmetric quantum Hall edge states and affine characters, arXiv:cond-mat/0409369 [cond-mat.mes-hall], 2004; Journal of Physics A: Mathematical and General 38.3 (2005): 617. - From _N. J. A. Sloane_, Apr 24 2014
C. Bessenrodt, On pairs of partitions with steadily decreasing parts, J. Combin. Theory, A 99 (2002), 162-174. MR1911463 (2003c:11133).
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London, 164 (1874), pp. 397-456, see pages 424 and 430.
Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230. But beware of typos in the g.f. on page 32. - _N. J. A. Sloane_, Apr 24 2014
B. Hemanthkumar and S. Chandankumar, New congruences modulo small powers of 2 for overpartitions into odd parts, Matematički Vesnik (2020).
M. D. Hirschhorn and J. A. Sellers, Arithmetic Properties of Overpartitions into Odd Parts, Annals of Combinatorics 10, no. 3 (2006), 353-367.
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.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69. See q-bar(n) on p. 66.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q-bar(n).
Vladimir Reshetnikov, A conjecture about algebraic values of (-q;-q)_oo/(q;q)_oo, Math Overflow Posting, Nov 23 2016, with proof supplied by Noam D. Elkies.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Cf. A000384, A007096, A029838, A080015, A098151, A103258, A103260, A108494, A215594, A261610, A261611.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2) +add(2*b(n-i*j, i-2), j=1..n/i)))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2014
# alternative program using expansion of f(x, x^3) / f(-x, -x^3):
with(gfun): series( add(x^(n*(2*n-1)), n = -8..8)/add((-1)^n*x^(n*(2*n-1)), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m )^(-1/8), {q, 0, n}]]; (* Michael Somos, Aug 03 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^(1/2), {q, 0, n}]; (* Michael Somos, Aug 03 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, q^2, q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[2*b[n - i*j, i - 2], {j, 1, n/i}]]];
a[n_] := b[n, n - 1 + Mod[n, 2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 05 2017, after Alois P. Heinz *)

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

a(n)=polcoeff(exp(2*sum(k=0,n\2,sigma(2*k+1)/(2*k+1)*x^(2*k+1))),n) /* Paul D. Hanna */

{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)), n))}; /* Michael Somos, Jul 07 2005 */

Expansion of f(q) / f(-q) in powers of q where f() is a Ramanujan theta function.
Expansion of (1 - k^2)^(-1/8) = k'^(-1/4) in powers of the nome q = exp(-Pi K'/K).
Expansion of eta(q^2)^3 / (eta(q^4) * eta(q)^2) in powers of q.
Euler transform of period 4 sequence [ 2, -1, 2, 0, ...].
(theta_3(q) / theta_4(q))^(1/2) = (phi(q) / phi(-q))^(1/2) = chi(q) / chi(-q) = psi(q) / psi(-q) = f(q) / f(-q) where phi{}, chi(), psi(), f() are Ramanujan theta functions.
G.f.: A(x) = exp( 2*sum_{n>=0} sigma(2*n+1)/(2*n+1)*x^(2*n+1) ). - Paul D. Hanna, Mar 01 2004
G.f. satisfies: A(-x) = 1/A(x), (A(x)+A(-x))/2 = A(x^2)*A(x^4)^2, A(x) = sqrt((A(x^2)^4+1)/2) + sqrt((A(x^2)^4-1)/2). - Paul D. Hanna, Mar 27 2004
Another g.f.: 1/product_{ k>= 1 } (1+x^(2*k))*(1-x^(2*k-1))^2. - Vladeta Jovovic, Mar 29 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v^3) * (v + 2*u^3) - u * (u^3 - v). - Michael Somos, Aug 03 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = (u^2 - v^2)^6 - 16 * u^2 * v^2 * (1 - u^8) * (1 - v^8). - Michael Somos, May 12 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A029838. - Michael Somos, Aug 03 2011
G.f.: (theta_3/theta_4)^(1/2) = ((Sum_{k in Z} x^(k^2))/(Sum_{k in Z} (-x)^(k^2)))^(1/2) = Product_{k>0} (1 - x^(4k-2))/((1 - x^(4k-1))(1 - x^(4k-3)))^2.
G.f.: Product_{ k >= 1 } (1 + x^(2*k-1))*(1 + x^k) = product_{ k >= 1 } (1 + x^(2*k-1))/(1 - x^(2*k-1)).
G.f.: 1 + 2*x / (1 - x) + 2*x^3 * (1 + x) / ((1 - x)*(1 - x^2)) + 2*x^6 * (1 + x)*(1 + x^2) / ((1 - x)*(1 - x^2)*(1 - x^3)) + ... [Glaisher 1876] - Michael Somos, Jun 20 2012
G.f.: 1 / (1 - 2*x / (1 + x - (x^2 - x^4) / (1 + x^3 - (x^3 - x^7) / (1 + x^5 - (x^4 - x^10) / (1 + x^7 - ...))))) [Glaisher 1878] - Michael Somos, Jun 24 2012
a(n) = (-1)^floor(n/2) * A080015(n) = (-1)^n * A108494(n). Convolution inverse is A108494. Convolution square is A007096.
Empirical : Sum_{n>=0} exp(-Pi)^n * a(n) = 2^(1/8). - Simon Plouffe, Feb 20 2011
Empirical : Sum_{n>=0} (-exp(-Pi))^n * a(n) = 1/2^(1/8). - Simon Plouffe, Feb 20 2011
a(n) ~ Pi * BesselI(1, Pi*sqrt(n/2)) / (4*sqrt(n)) ~ exp(Pi*sqrt(n/2)) / (2^(9/4) * n^(3/4)) * (1 - 3/(4*Pi*(sqrt(2*n))) - 15/(64*Pi^2*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
Simon Plouffe's empirical observations are true. Furthermore, for every positive rational p, Sum_{n>=0} exp(-Pi*sqrt(p))^n * a(n) = 1/(Sum_{n>=0} (-exp(-Pi*sqrt(p)))^n * a(n)) is an algebraic number (see the MathOverflow link). - Vladimir Reshetnikov, Nov 23 2016
G.f.: f(x,x^3)/f(-x,-x^3) = ( Sum_{n = -oo..oo} x^(n*(2*n-1)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(2*n-1)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. - Peter Bala, Feb 05 2021
G.f. A(q) = (-lambda(-q)/lambda(q))^(1/8), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus. - Peter Bala, Sep 26 2023
Recurrence: a(n) = c(n) + Sum_{k = 1..floor((-1 + sqrt(1 + 8*n))/2)} (-1)^(1 + k*(k+1)/2) * a(n - k*(k+1)/2), where c(n) = 1 if n is a triangular number, otherwise c(n) = 0. See A010054. - Peter Bala, Jun 08 2025

Definition simplified by N. J. A. Sloane, Apr 24 2014

A029839 McKay-Thompson series of class 16B for the Monster group.

Original entry on oeis.org

1, 2, -1, -2, 3, 2, -4, -4, 5, 8, -8, -10, 11, 12, -15, -18, 22, 26, -29, -34, 38, 42, -51, -56, 66, 78, -85, -98, 109, 120, -139, -156, 176, 202, -222, -250, 279, 306, -346, -384, 429, 482, -530, -590, 650, 714, -797, -876, 972, 1080, -1180, -1304, 1431, 1562, -1728, -1892, 2078, 2290, -2496

Offset: 0

N. J. A. Sloane

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

In [Klein and Fricke 1890], the g.f. A(q)/2 is denoted by mu. On page 613 special values given are mu(i infinity) = infinity, mu(0) = 1, mu(2) = -1 and on page 615 properties given are mu(omega+1) = -i mu(omega), mu(-1/omega) = (mu(omega)+1)/(mu(omega)-1). - Michael Somos, Nov 09 2014

			G.f. = 1 + 2*x - x^2 - 2*x^3 + 3*x^4 + 2*x^5 - 4*x^6 - 4*x^7 + 5*x^8 + 8*x^9 + ...
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
F. Klein and R. Fricke, Vorlesungen über die theorie der elliptischen modulfunctionen, Teubner, Leipzig, 1890, Vol. 1, see pp. 613, 615, 675.
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to groups
Index entries for McKay-Thompson series for Monster simple group

Cf. A079006, A082304, A029838.

Product_{m>=1} ((1+q^(2*m-1))/(1+q^(2*m)))^b: this sequence (b=1), A029839 (b=2), A029840 (b=3), A029841 (b=4), A029842 (b=5), A029843 (b=6), A029844 (b=7).

a[0] = 1; a[n_] := Module[{A, m}, If[n < 0, 0, A = 1; m = 1; While[m <= n, m *= 2; A = A /. x -> x^2; A = Sqrt[A + 4*x/A]]; SeriesCoefficient[A, {x, 0, n}]]]; Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Mar 12 2014, after PARI *)
a[ n_] := SeriesCoefficient[ 2 q^(1/4) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]^4) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from PARI *)

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

{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 + 4*x/A)); polcoeff(A, n))};

Expansion of q times normalized Hauptmodul for Gamma(4) in powers of q^4.
Expansion of q^(1/4) * eta(q^2)^6 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [2, -4, 2, 0, ...].
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 4*x / A(x^2). - Michael Somos, Mar 08 2004
G.f.: Product_{k>0} ((1 + x^(2*k-1)) / (1 + x^(2*k)))^2.
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^2 - u^2*v. - Michael Somos, May 14 2004
Given g.f. A(x), then B(q) = A(q^4) / (2*q) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Oct 04 2006
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (u6 + u2)^2 - u1*u2*u3*u6. - Michael Somos, Oct 04 2006
Convolution inverse of A079006.
Expansion of q^(1/4) * 2 / k(q)^(1/2) in powers of Jacobi nome q where k() is the elliptic modulus.
Expansion of q^(1/2) * 2 * (1 + k'(q)) / k(q) in powers of q^2. - Michael Somos, Nov 09 2014
Expansion of phi(x) / psi(x^2) = phi(x)^2 / psi(x)^2 = psi(x)^2 / psi(x^2)^2 = phi(-x^2)^2 / psi(-x)^2 = chi(-x^2)^4 / chi(-x)^2 = chi(x)^2 * chi(-x^2)^2 = chi(x)^4 * chi(-x)^2 = f(x)^2 / f(-x^4)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of continued fraction 1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)) in powers of x^4. - Michael Somos, Apr 27 2008
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 the g.f. for A007096.
a(n) = (-1)^n * A082304(n). Convolution square is A029841. - Michael Somos, Jul 05 2014
From Peter Bala, Jan 09 2021: (Start)
A(q) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+1)) / Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)).
A(q) = ( 1 + q/(1 + (q + q^2)/(1 + q^3/(1 + (q^2 + q^4)/(1 + q^5/(1 + ... ))))) )^2. See Agarwal, p. 285.
A(q) = B(q)^2, where B(q) is the g.f. of A029838. (End)
abs(a(n)) ~ exp(Pi*sqrt(n)/2) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

Additional comments from Michael Somos, Jul 11 2002

A022577 Expansion of Product_{m>=1} (1+x^m)^12.

Original entry on oeis.org

1, 12, 78, 376, 1509, 5316, 16966, 50088, 138738, 364284, 913824, 2203368, 5130999, 11585208, 25444278, 54504160, 114133296, 234091152, 471062830, 931388232, 1811754522, 3471186596, 6556994502, 12222818640, 22502406793, 40944396120, 73680871326, 131211105208, 231355524048, 404110659732

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 12*x + 78*x^2 + 376*x^3 + 1509*x^4 + 5316*x^5 + 16966*x^6 + ...
G.f. = q + 12*q^3 + 78*q^5 + 376*q^7 + 1509*q^9 + 5316*q^11 + 16966*q^13 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Column k=12 of A286335.

Cf. A000122, A000700, A007096, A007249, A010054, A121373, A124863.

Coefficients(&*[(1+x^m)^12:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

N:= 50:
G:= mul(1+x^m,m=1..N+1)^12:
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Feb 26 2018

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q)^(1/2) / (1 - m), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (m / 16 / q) /(1-m)^(1/2), {q, 0, 2 n}]]; (* Michael Somos, Jul 22 2011 *)
CoefficientList[QPochhammer[-1, q]^12/4096+O[q]^30, q] (* Jean-François Alcover, Nov 27 2015 *)
With[{nmax=50}, CoefficientList[Series[Product[(1+q^k)^12, {k,1,nmax}], {q, 0, nmax}],q]] (* G. C. Greubel, Feb 25 2018 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 + x^k, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Jul 16 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^12, n))}; /* Michael Somos, Jul 16 2005 */

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^12)) \\ G. C. Greubel, Feb 25 2018

G.f.: Product_{k>=1} ( 1 + x^k )^12.
Expansion of chi(-x)^-12 in powers of x where chi() is a Ramanujan theta function.
Expansion of k^2 / (16 * q * k') in powers of q^2. - Michael Somos, Jul 22 2011
Expansion of q^(-1/2) * (k/4) / (1 - k^2) in powers of q. - Michael Somos, Jul 16 2005
Expansion of q^(-1/2) * (eta(q^2) / eta(q))^12 in powers of q. - Michael Somos, Jul 16 2005
Euler transform of period 2 sequence [12, 0, ...]. - Michael Somos, Jul 16 2005
Given g.f. A(x), then B(q) = (q * A(q^2))^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (4096*u*v + 48*u + 1)*v - u^2 . - Michael Somos, Jul 16 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/64 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007249. - Michael Somos, Jul 22 2011
A124863(n) = (-1)^n * a(n). A007096(4*n + 2) = 8 * a(n). Convolution inverse of A007249.
a(n) ~ exp(2 * Pi * sqrt(n)) / (128 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015
a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

More terms added by G. C. Greubel, Feb 25 2018

A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369

Offset: 0

Michael Somos, Jan 29 2006

nonn

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

Andrews (1987) refers to this sequence as p(S, n) where S is the set in equation (1) on page 437.

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + 15*x^9 + ...
a(5) = 4 since 5 = 4 + 1 = 3 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 4 ways.
a(6) = 6 since 6 = 5 + 1 = 4 + 1 + 1 = 3 + 3 = 3 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 6 ways.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..120 from Reinhard Zumkeller)
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 32. MR0858826 (88b:11063).
G. E. Andrews, Unsolved Problems: Further Problems on Partitions, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157, December 2015, Pages 223-232.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080054, A187154, A208851, A208856, A218171.

a115671 = p [x | x <- [0..], (mod x 32) `notElem` [0,2,12,14,16,18,20,30]]
   where p _          0 = 1
         p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 03 2012

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 / QPochhammer[ q]^2 / QPochhammer[ q^4] + 1) / 2, {q, 0, n}]; (* Michael Somos, Nov 09 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

Expansion of (f(q) / f(-q) + 1) / 2 in powers of q where f() is a Ramanujan theta function.
Expansion of f(q^6, q^10) / f(-q, -q^3) = (f(q^22, q^26) - q^2 * f(q^10, q^38)) / f(-q, -q^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(x) = (2*A(x) - 1)^2 = g.f. A007096 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = 1 + u^2 - 2 * u * v^2.
G.f. (1 + sqrt( theta_3(x) / theta_4(x))) / 2 = (Sum_{k} x^(8*k^2 - 2*k)) / (Sum_{k} (-x)^(2*k^2 - k)) = (Sum_{k} x^(24*n^2 + 2*n) - x^(24*n^2 + 14*n + 2)) / (Product_{k>0} 1 - x^k).
2 * a(n) = A080054(n) unless n = 0. a(2*n + 2) = A208851(n). a(2*n + 1) = A187154(n). a(n + 1) = A208856(n).

A123655 Expansion of q * psi(q^8) / phi(-q) in powers of q where psi(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 40, 64, 101, 156, 236, 352, 518, 752, 1080, 1536, 2162, 3018, 4180, 5744, 7840, 10632, 14328, 19200, 25591, 33932, 44776, 58816, 76918, 100176, 129952, 167936, 216240, 277476, 354864, 452392, 574958, 728568, 920600, 1160064

Offset: 1

Michael Somos, Oct 04 2006

nonn

Ramanujan theta functions: phi(q) (A000122), psi(q) (A010054).

Number 12 of the 14 eta-quotients listed in Table 2 of Moy 2013. - Michael Somos, Sep 19 2013

			G.f. = q + 2*q^2 + 4*q^3 + 8*q^4 + 14*q^5 + 24*q^6 + 40*q^7 + 64*q^8 + 101*q^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
Richard Moy, Congruences among power series coefficients of modular forms, arXiv:1309.4320
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A007096, A185338, A093160, A107035, A185338.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^4] / EllipticTheta[ 4, 0, q] / 2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)), n))};

Expansion of eta(q^2) * eta(q^16)^2 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 16 sequence [ 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 2*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/8 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A185338.
a(n) is odd iff n is an odd square. If n>2 is a power of 2 then the highest power of 2 dividing a(n) is (n/2)^3. - Michael Somos, Feb 18 2007
4 * a(n) = A007096(n) unless n=0. -(-1)^n * a(n) = A208605(n). Convolution inverse of A185338.
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k)) * (1 + x^(8*k))^2. Michael Somos, Sep 19 2013
a(2*n) = 2 * A107035(n). a(2*n + 1) = A093160(n). - Michael Somos, Sep 19 2013
a(n) ~ exp(sqrt(n)*Pi) / (2^(9/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A208933 Expansion of phi(q^4) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 48, 80, 128, 202, 312, 472, 704, 1036, 1504, 2160, 3072, 4324, 6036, 8360, 11488, 15680, 21264, 28656, 38400, 51182, 67864, 89552, 117632, 153836, 200352, 259904, 335872, 432480, 554952, 709728, 904784, 1149916, 1457136, 1841200, 2320128

Offset: 0

Michael Somos, Mar 13 2012

nonn

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

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 16*q^4 + 28*q^5 + 48*q^6 + 80*q^7 + 128*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A093160, A112128, A123655, A131126, A208603, A208604.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)
nmax=60; CoefficientList[Series[Product[(1-x^(2*k)) * (1-x^(8*k))^5 / ((1-x^k)^2 * (1-x^(4*k))^2 * (1-x^(16*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^8 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^2, n))};

Expansion of eta(q^2) * eta(q^8)^5 / (eta(q) * eta(q^4) * eta(q^16))^2 in powers of q.
Euler transform of period 16 sequence [ 2, 1, 2, 3, 2, 1, 2, -2, 2, 1, 2, 3, 2, 1, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208603.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2*u - 1) * (2*v^2 - 2*v + 1) - u^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = 4 * u * (u - 1) * (2*u - 1) * v * (v - 1) * (2*v - 1) - (u - v)^4.
(-1)^n * a(n) = A112128(n). a(n) = 2 * A123655(n) unless n=0. 2 * a(n) = A007096(n) unless n=0. a(2*n) = A131126(n). a(2*n + 1) = 2 * A093160(n). Convolution inverse of A208604.
G.f.: (Sum_{k in Z} x^(4 * k^2)) / (Sum_{k in Z} (-1)^k * x^(k^2)) = theta_3(x^4) / theta_3(-x).
G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + (-x)^k) * (1 + x^(8*k)))^2.
a(n) ~ exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

A097243 Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.

Original entry on oeis.org

1, 32, 256, 1408, 6144, 22976, 76800, 235264, 671744, 1809568, 4640256, 11404416, 27009024, 61905088, 137803776, 298806528, 632684544, 1310891584, 2662655232, 5310231424, 10412576768, 20098970624, 38231811072, 71734039808, 132875747328, 243175399136

Offset: 0

Michael Somos, Aug 02 2004

nonn

Expansion of a q-series used in construction of j(tau) to j(2tau) iteration.

			G.f. = 1 + 32*x + 256*x^2 + 1408*x^3 + 6144*x^4 + 22976*x^5 + 76800*x^6 + ...

H. Cohn, Introduction to the construction of class fields, Cambridge 1985, p. 191

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

Cf. A007248, A007096, A014969, A092877, A139820, A189925, A212318, A232358.

a[ n_] := SeriesCoefficient[ 1 + 32 x (QPochhammer[ x^4] / QPochhammer[ x])^8, {x, 0, n}]; (* Michael Somos, Dec 15 2016 *)

{a(n) = my(A); if( n<0, 0, A = x^n * O(x); polcoeff( 1 + 32 * x * (eta(x^4 + A) / eta(x + A))^8, n))};

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+3)^2 - 8*(u+1)*v^2.
a(n) = 32*A092877(n), if n>0. a(n) = A007096(4*n).
a(n) = A014969(2*n) = A139820(2*n) = A189925(4*n) = A212318(4*n) = A232358(4*n). - Michael Somos, Dec 15 2016
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007248. - Michael Somos, Dec 15 2016
a(n) ~ exp(2*Pi*sqrt(n))/(16*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

A103261 Number of partitions of 2n into parts with 10 types c^1 c^2...C^10 of each part. The even parts appear with multiplicity 1 for each type . The odd parts occur with multiplicity 2 for each part.

Original entry on oeis.org

1, 20, 200, 1360, 7200, 32024, 125280, 443680, 1450240, 4435940, 12827888, 35346800, 93377920, 237675640, 585229760, 1398704736, 3253934080, 7386124520, 16392493800, 35634450320, 75992326592, 159199081600, 328027789600

Offset: 0

Noureddine Chair, Feb 16 2005

nonn

This is also Sequence(A080054)^(10) or sequence(A007096)^(5).

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

			a(2)=200 because we have 10 types of 4, 45 ways of writing 4 in terms of ten of 2's only or ten of 11's only and 100 ways of writing 2's combined with 11's so the total number of ways of writing 4 is 200.

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A261648 (j=5), A014970 (j=6), A014972 (j=8).

series(product(((1+x^k)*(1-x^(2*k)))^(10)/((1-x^k)*(1+x^(2*k)))^(10),k=1..100),x=0,100);

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^10,{k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Aug 28 2015 *)

G.f.:(theta_4(0, x^2)/theta_4(0, x))^10= (theta_3(0, x)/theta_4(0, x))^5.

a(n) ~ exp(Pi*sqrt(5*n)) * 5^(1/4) / (64 * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

Example corrected by Vaclav Kotesovec, Sep 01 2015

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

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A080054 G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).

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A029839 McKay-Thompson series of class 16B for the Monster group.

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A022577 Expansion of Product_{m>=1} (1+x^m)^12.

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A115671 Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).

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A123655 Expansion of q * psi(q^8) / phi(-q) in powers of q where psi(), phi() are Ramanujan theta functions.

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A208933 Expansion of phi(q^4) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

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A261647 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^3.