A098151 -id:A098151 - 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

A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 12, 28, 60, 120, 228, 416, 732, 1252, 2088, 3408, 5460, 8600, 13344, 20424, 30876, 46152, 68268, 100016, 145224, 209120, 298800, 423840, 597108, 835804, 1162824, 1608508, 2212896, 3028632, 4124664, 5590976, 7544604, 10137264, 13565016

Offset: 0

Michael Somos, Mar 05 2011

nonn

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

			G.f. = 1 + 4*q + 12*q^2 + 28*q^3 + 60*q^4 + 120*q^5 + 228*q^6 + 416*q^7 + 732*q^8 + ...

G. C. Greubel, 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

Cf. A058487, A098151, A187100, A217771.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^2 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
nmax = 50; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k))^2 / ((1-x^k)^2 * (1-x^(6*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

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

Euler transform of period 6 sequence [ 4, 2, 0, 2, 4, 0, ...].
Expansion of (eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)))^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/3)  g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A058487.
Convolution square of A098151. a(n) = 4 * A187100(n) unless n=0.
Convolution inverse of A217771. - Michael Somos, Sep 05 2015
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -6, 10, -16, 24, -36, 52, -74, 104, -144, 198, -268, 360, -480, 634, -832, 1084, -1404, 1808, -2316, 2952, -3744, 4728, -5946, 7448, -9294, 11556, -14320, 17688, -21780, 26740, -32736, 39968, -48672, 59122, -71644, 86616, -104484, 125768, -151072

Offset: 0

Michael Somos, Aug 06 2007

sign

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

			G.f. = 1 - 2*x + 4*x^2 - 6*x^3 + 10*x^4 - 16*x^5 + 24*x^6 - 36*x^7 + 52*x^8 + ...

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 p. 15.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A098151, A139007, A139136, A139137, A261320.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Apr 04 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q^3] QPochhammer[ -q^2, -q^3] / (QPochhammer[ -q, -q^3] QPochhammer[ q^2, -q^3]), {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x * O(x^n)) / sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^5 / ( eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -2, 3, 0, 1, -2, 0, -2, 1, 0, 3, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v+u) * (v-u) + (1 - u*v) * (1 - 3*u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3*u*v^2 * (1 - u*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139007. - Michael Somos, Apr 04 2015
G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2).
G.f.: Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)). - Michael Somos, Apr 04 2015
a(n) = (-1)^n * A098151(n).
Convolution inverse of A139137. Convolution square is A261320. - Michael Somos, Aug 14 2015
Expansion of f(-q, q^2) / f(q, -q^2) in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 02 2015
a(n) = A139136(3*n) = A139137(3*n). - Michael Somos, Nov 02 2015
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A103257 Number of partitions of 2n free of multiples of 5. All odd parts occur with multiplicity 2 or 4. the even parts occur at most twice.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 28, 40, 54, 72, 96, 126, 164, 212, 272, 346, 436, 548, 684, 850, 1052, 1296, 1588, 1940, 2362, 2864, 3462, 4172, 5012, 6004, 7172, 8548, 10160, 12048, 14256, 16830, 19828, 23312, 27356, 32040

Offset: 0

Noureddine Chair, Jan 27 2005

nonn

Convolution of A261796 and A261797. - Vaclav Kotesovec, Sep 01 2015

			E.g. a(5) = 14 because 10 can be written as 8+2 = 8+1+1 = 6+4 = 6+2+2 = 6+2+1+1 = 6+1+1+1+1 = 4+4+2 = 4+4+1+1 = 4+3+3 = 4+2+2+1+1 = 4+2+1+1+1+1 = 3+3+2+2 = 3+3+2+1+1 = 3+3+1+1+1+1.

Robert Israel, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A098151, A261796, A261797.

series(product(((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);

nmax = 50; CoefficientList[Series[Product[((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

G.f.: (theta_4(0, x^3)*theta_4(0, x^5))/theta_4(0, x).
G.f.: (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) where E(k) = Product_{n>=1} 1-q^(k*n). - Joerg Arndt, Sep 01 2015
a(n) ~ exp(Pi*sqrt(7*n/15)) / sqrt(15*n). - Vaclav Kotesovec, Sep 01 2015

Example corrected by Vaclav Kotesovec, Sep 01 2015

Maple program fixed by Vaclav Kotesovec, Sep 01 2015

A122792 Expansion of eta(q^2)^2/(eta(q)eta(q^3)) in powers of q.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 2, 0, 6, 4, 0, 10, 6, 0, 16, 9, 0, 24, 14, 0, 36, 20, 0, 52, 29, 0, 74, 42, 0, 104, 58, 0, 144, 80, 0, 198, 110, 0, 268, 148, 0, 360, 198, 0, 480, 264, 0, 634, 347, 0, 832, 454, 0, 1084, 592, 0, 1404, 764, 0, 1808, 982, 0, 2316, 1257, 0, 2952, 1598, 0

Offset: 0

Michael Somos, Sep 11 2006

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A098151(n)=a(3n). A097197(n)=a(3n+1).

Cf. A293306.

QP = QPochhammer; s = QP[q^2]^2/(QP[q]*QP[q^3]) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

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

Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, 0, ...].
G.f.: Product_{k>0} (1-x^k)^2/(1+x^k+x^(2k)). a(3n+2)=0.
G.f.: Product_{i>0} 1/(1 + Sum_{j>0} (-1)^j*j*q^(j*i)). - Seiichi Manyama, Oct 08 2017

A256626 Expansion of psi(x) / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 1, 1, 0, -2, -1, 0, 2, 2, 0, -2, -3, 0, 3, 3, 0, -4, -4, 0, 5, 6, 0, -6, -7, 0, 7, 8, 0, -10, -10, 0, 13, 13, 0, -14, -16, 0, 17, 18, 0, -22, -22, 0, 26, 28, 0, -30, -33, 0, 36, 38, 0, -44, -45, 0, 52, 55, 0, -60, -65, 0, 70, 74, 0, -84

Offset: 0

Michael Somos, Apr 05 2015

sign

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

			G.f. = 1 + x - x^4 + x^6 + x^7 - 2*x^9 - x^10 + 2*x^12 + 2*x^13 - 2*x^15 + ...
G.f. = 1/q + q^3 - q^15 + q^23 + q^27 - 2*q^35 - q^39 + 2*q^47 + 2*q^51 + ...

G. C. Greubel, 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

Cf. A058487, A098151, A101195, A128633, A135211.

a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}];

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

Expansion of f(-x^2, -x^4) / f(-x, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.
Expansion of q^(1/4) * eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of
Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, 0, ...].
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 - (v^2 - 1) * (u^4 - v^2).
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u*v - 1)^3 - (v^4 - 1).
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (v^2 + w^2) - v*w * (3 + v^2).
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) = 3*u2 + u1*u3*u6 - u2^2*u6 - u1*u2*u3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A098151.
G.f.: Product_{k>0} (1 + x^k + x^(2*k))^-1 * (1 - x^k + x^(2*k))^-2.
Convolution inverse is A101195. Convolution square is A058487. Convolution 4th power is A128633.
a(n) = (-1)^n * A135211(n). a(3*n + 2) = 0.

A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596, 421844, 513834

Offset: 0

Noureddine Chair, Dec 20 2004; revised Jan 05 2005

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).
There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos, Oct 27 2008

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Cristina Ballantine, Mircea Merca, Jacobi's Four and Eight Squares Theorems and Partitions into Distinct Parts, Mediterranean Journal of Mathematics (2019) Vol. 16, No. 2, 26.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Ramanujan Theta Functions

Cf. A015128, A098151, A080054.

series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))),k=1..100),x=0,100);

nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(2/(QPochhammer[x] QPochhammer[-1, -x]) + O[x]^45)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 10 2005 */

Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos, Feb 10 2005
G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).
Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Oct 27 2008
Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Oct 27 2008
a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Aug 30 2015
G.f.: 2/((x; x)inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016
Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - Michael Somos, Nov 22 2016

A102346 Number of partitions of 2n in which odd parts and multiples of 3 and 5 occur with even multiplicities. There is no restriction on the other even parts.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 769, 1042, 1401, 1871, 2482, 3273, 4291, 5596, 7261, 9378, 12057, 15437, 19684, 25005, 31648, 39919, 50184, 62890, 78573, 97883, 121597, 150653, 186169, 229487, 282204, 346230, 423831, 517706

Offset: 0

Noureddine Chair, Feb 21 2005

nonn

			a(5) = 19: [8,2], [8,1,1], [5,5], [4,4,2], [4,4,1,1], [4,2,2,2], [4,2,2,1,1], [4,2,1,1,1,1], [4,3,3], [3,3,2,2], [3,3,2,1,1], [3,3,1,1,1,1], [4,1,1,1,1,1,1], [2,2,2,2,2], [2,2,2,2,1,1], [2,2,2,1,1,1,1], [2,2,1,1,1,1,1,1], [2,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Cf. A098151.

nmax = 50; CoefficientList[Series[Product[(1+x^k)*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)*E(5)*E(30)) / (E(1)^2*E(6)*E(10)*E(15)) ) \\ Joerg Arndt, Sep 01 2015

G.f.: Product((1+x^k)*(1+x^(15*k))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), k>=1).
a(n) ~ exp(Pi*sqrt(38*n/5)/3) * sqrt(19) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015
G.f.: (E(2)*E(3)*E(5)*E(30)) / (E(1)^2*E(6)*E(10)*E(15)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015

Corrected by Vladeta Jovovic, Feb 21 2005

Offset and example corrected by Vaclav Kotesovec, Sep 01 2015

A215594 Expansion of f(-x, -x^4) / f(x, x^4) in powers of x where f(,) is Ramanujan's two-variable theta function.

Original entry on oeis.org

1, -2, 2, -2, 0, 2, -4, 6, -4, 0, 6, -12, 14, -10, 0, 14, -26, 30, -22, 0, 28, -52, 60, -42, 0, 54, -100, 112, -78, 0, 100, -180, 202, -140, 0, 174, -314, 350, -240, 0, 296, -532, 588, -402, 0, 492, -876, 966, -658, 0, 794, -1412, 1550, -1050, 0, 1260, -2232

Offset: 0

Michael Somos, Aug 16 2012

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

Let A(q) denote the g.f. of this sequence. Let m be a nonzero integer. The simple continued fraction expansions of the real numbers A(1/(2*m)) and A(1/(2*m+1)) may be predictable. For a given positive integer n, the sequence of the n-th partial denominators of the continued fractions are conjecturally polynomial or quasi-polynomial in m for m sufficiently large. An example is given below. Cf. A080054 and A098151. - Peter Bala, Jun 10 2025

			1 - 2*x + 2*x^2 - 2*x^3 + 2*x^5 - 4*x^6 + 6*x^7 - 4*x^8 + 6*x^10 - 12*x^11 + ...
From _Peter Bala_, Jun 10 2025: (Start)
G.f.: A(q) = f(-q, -q^4) / f(q, q^4).
Simple continued fraction expansions of A(1/(2*m)):
  m = 2 [0; 1   1  2    7  1  1   1  10   1  2   12     82  1  6  1   48 ...]
  m = 3 [0; 1   2  2   17  1  1   2  14   2  2   24    318  1  1  1    1 ...]
  m = 4 [0; 1   3  2   31  1  1   3  18   3  2   40    810  2  7  2  161 ...]
  m = 5 [0; 1   4  2   49  1  1   4  22   4  2   60   1654  2  1  1    1 ...]
  m = 6 [0; 1   5  2   71  1  1   5  26   5  2   84   2946  3  7  1    1 ...]
  m = 7 [0; 1   6  2   97  1  1   6  30   6  2  112   4782  3  1  1    1 ...]
  m = 8 [0; 1   7  2  127  1  1   7  34   7  2  144   7258  4  7  1    2 ...]
  m = 9 [0; 1   8  2  161  1  1   8  38   8  2  180  10470  4  1  1    1 ...]
 m = 10 [0; 1   9  2  199  1  1   9  42   9  2  220  14514  5  7  1    3 ...]
 m = 11 [0; 1  10  2  241  1  1  10  46  10  2  264  19486  5  1  1    1 ...]
 m = 12 [0; 1  11  2  287  1  1  11  50  11  2  312  25482  6  7  1    4 ...]
 ...
The sequence of the 4th partial denominators [7, 17, 31, 49, ...] appears to be given by the polynomial 2*m^2 - 1 for m >= 2.
The sequence of the 11th partial denominators [12, 24, 40, 60, ...] appears to be given by the polynomial 2*(m^2 + m) for m >= 2.
The sequence of the 12th partial denominators [82, 318, 810, 1654, ...] appears to be given by the polynomial 2*(8*m^3 - 8*m^2 + 6*m - 3) for m >= 2.
The sequence of the 16th partial denominators appears to become quasi-polynomial in m for m >= 5. (End)

G. C. Greubel, 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

Cf. A080054, A098151.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A215594[n_] := SeriesCoefficient[f[-x, -x^4]/f[x, x^4], {x, 0, n}]; Table[A215594[n], {n,0,50}] (* G. C. Greubel, Jun 18 2017 *)

{a(n) = local(A, s); if( n<0, 0, A = x * O(x^n); s = sqrtint( 40*n + 9); polcoeff( sum( k=(-s + 6)\10, (s - 3)\10, (-1)^k *  x^((5*k + 3)*k/2), A) / sum( k=(-s + 6)\10, (s - 3)\10,  x^((5*k + 3)*k/2), A), n))}

Euler transform of period 10 sequence [ -2, 1, 0, -2, 0, -2, 0, 1, -2, 0, ...].

a(5*n + 4) = 0.

cp's OEIS Frontend

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

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A186924 Expansion of (phi(-q^3) / phi(-q))^2 in powers of q where phi is a Ramanujan theta function.

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A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.

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A103257 Number of partitions of 2n free of multiples of 5. All odd parts occur with multiplicity 2 or 4. the even parts occur at most twice.

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A122792 Expansion of eta(q^2)^2/(eta(q)eta(q^3)) in powers of q.

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A256626 Expansion of psi(x) / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

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A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

A261796 Expansion of Product_{k>=1} (1+x^k)/((1+x^(3k))(1+x^(5*k))).