A121373 -id:A121373 - 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

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

A053692 Number of self-conjugate 4-core partitions of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 3, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0

Offset: 0

James Sellers, Feb 14 2000

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

Also the number of positive odd solutions to equation x^2 + 4*y^2 = 8*n + 5. - Seiichi Manyama, May 28 2017

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^10 + x^12 + x^13 + x^14 + 2*x^15 + ...
G.f. = q^5 + q^13 + q^29 + q^37 + q^45 + q^53 + q^61 + 2*q^85 + q^101 + q^109 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Christopher R. H. Hanusa and Rishi Nath, The number of self-conjugate core partitions, arxiv:1201.6629 [math.NT], 2012. See Table 1 p. 15.
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A053693.

A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* Michael Somos, Jun 21 2015 */;

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x (1/2)] EllipticTheta[ 2, 0, x^2] / (4 x^(5/8)), {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^8]^2, {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^2, x^4] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2]^2 - 2 EllipticTheta[ 2, Pi/4, q^2]^2) / 16, {q, 0, 8 n + 5}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := If[ n < 0, 0, Sum[ (-1)^Quotient[d, 2], {d, Divisors[ 8 n + 5]}] / 2]; (* Michael Somos, Jun 21 2015 *)

{a(n) = my(A); if( n<0, 0, A = sum( k=0, ceil( sqrtint(8*n + 1)/2), x^((k^2 + k)/2), x * O(x^n)); polcoeff( A * subst(A + x * O(x^(n\4)), x, x^4), n))}; /* Michael Somos, Nov 03 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}; /* Michael Somos, Apr 28 2003 */

{a(n) = if( n<0, 0, sumdiv( 8*n + 5, d, (-1)^(d\2)) / 2)}; /* Michael Somos, Jun 21 2015*/

Expansion of psi(x) * psi(x^4) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 03 2005
Expansion of chi(x) * f(-x^8)^2 in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 24 2012
Expansion of f(x, x^7) * f(x^3, x^5) = f(x, x^3) * f(x^4, x^12) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 21 2015
Expansion of (psi(x)^2 - psi(-x)^2) / (4*x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Jun 21 2015
Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q. - Michael Somos, Apr 28 2003
Euler transform of period 8 sequence [ 1, -1, 1, 0, 1, -1, 1, -2, ...]. - Michael Somos, Apr 28 2003
a(n) = 1/2 * b(8*n + 5), where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jul 24 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246950.
G.f.: Sum_{k in Z} x^k / (1 - x^(8*k + 5)). - Michael Somos, Nov 03 2005
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 - x^(2*k - 1)). - Michael Somos, Jun 21 2015
G.f.: Product_{i>=1} (1-x^(8*i))^2*(1-x^(4*i-2))/(1-x^(2*i-1)).
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = a(n). 2 * a(n) = A008441(2*n + 1).

A098151 Number of partitions of 2*n with no part divisible by 3 and all odd parts occurring with even multiplicities.

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, 181104, 216684

Offset: 0

Noureddine Chair, Aug 29 2004

There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015
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. - Peter Bala, Jun 09 2025

			a(4)=10 because 8 = 4+4 = 4+2+2=2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 4+2+1+1 = 4+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...
From _Peter Bala_, Jun 09 2025: (Start)
G.f.: A(q) = f(q, q^2) / f(-q, -q^2).
Simple continued fraction expansions of A(1/(2*m)):
m =  2  [1;  1   9  1    5    8    45   4  1  2  1  1  1  3  3   2  2 ...]
m =  3  [1;  2  13  1   14   12   133   8  1  1  7  2  1  2  2   1  1 ...]
m =  4  [1;  3  17  1   27   16   297  12  2  2  1  1  1  2  2   2  2 ...]
m =  5  [1;  4  21  1   44   20   561  16  2  1  7  3  3  2  2  25  8 ...]
m =  6  [1;  5  25  1   65   24   949  20  3  2  1  1  1  3  4   2  1 ...]
m =  7  [1;  6  29  1   90   28  1485  24  3  1  7  4  5  2  1   1  6 ...]
m =  8  [1;  7  33  1  119   32  2193  28  4  2  1  1  1  4  6   2  1 ...]
m =  9  [1;  8  37  1  152   36  3097  32  4  1  7  5  7  2  1   1  3 ...]
m = 10  [1;  9  41  1  189   40  4221  36  5  2  1  1  1  5  8   2  1 ...]
...
The sequence of the 4th partial denominators [5, 14, 27, 44, ...] appears to be given by the polynomial (2*m + 1)*(m - 1) for m >= 2.
The sequence of the 6th partial denominators [45, 133, 297, 561, ...] appears to be given by the polynomial (2*m + 1)*(2*m^2 + 1) for m >= 2. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
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, 14-16.
Byungchan Kim and Eunmi Kim, Partitions weighted by the number of two types of parts, Bull. Korean Math. Soc. (2024) Vol. 61, No. 6, 1677-1684. See p. 1679.
Jeremy Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory. 1 (2005) 215-224
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A015128, A001318, A080054, A080995, A010815, A103260, A215594.

series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
# alternative program using expansion of f(q, q^2) / f(-q, -q^2):
with(gfun): series( add(x^(n*(3*n-1)/2),n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(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)), n))} /* Michael Somos, Dec 04 2004 */

Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012
Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012
Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
G.f. = (Sum_{n = -oo..oo} (-1)^n*q^(3*n^2)) / (Sum_{n = -oo..oo} (-1)^n*q^(n^2)). - N. J. A. Sloane, Aug 09 2016
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004
Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004
Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)
a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017
Convolution of A000726 and A003105. - R. J. Mathar, Nov 17 2017
From Peter Bala, Jun 09 2025: (Start)
G.f.: A(q) = Sum_{n = -oo..oo} q^(n*(3*n+1)/2) / Sum_{n = -oo..oo} (-1)^n * q^(n*(3*n+1)/2).
Recurrences:
a(n) - a(n-1) - a(n-2) + a(n-5) + a(n-7) - a(n-12) - a(n-15) + + - - ... = f(n), where [0, 1, 2, 5, 7, 12, 15, ...] is the sequence of generalized pentagonal numbers A001318, a(n) is set equal to 0 for negative n and f(n) = 1 if n is a generalized pentagonal number, otherwise f(n) = 0 (see A080995). Compare with the recurrence for the partition function p(n) = A000041(n).
a(n) - 2*a(n-1) + 2*a(n-4) - 2*a(n-9) + 2*a(n-16) - 2*a(n-25) + - ... = g(n), where g(n) = 2*(-1)^k if n is of the form 3*(k^2), otherwise g(n) = 0. (End)

A122135 Expansion of f(x, -x^4) / phi(-x^2) in powers of x where f(, ) and phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882, 14594

Offset: 0

Michael Somos, Aug 21 2006

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of integer partitions y of n such that y_i > y_{i+1} for all even i. For example, the a(1) = 1 through a(9) = 12 partitions are:
  (1)  (2)   (3)   (4)   (5)    (6)     (7)     (8)     (9)
       (11)  (21)  (22)  (32)   (33)    (43)    (44)    (54)
                   (31)  (41)   (42)    (52)    (53)    (63)
                         (221)  (51)    (61)    (62)    (72)
                                (321)   (331)   (71)    (81)
                                (2211)  (421)   (332)   (432)
                                        (3211)  (431)   (441)
                                                (521)   (531)
                                                (3311)  (621)
                                                (4211)  (3321)
                                                        (4311)
                                                        (5211)
The even-length case appears to be A122134.
The odd-length case is A351595.
The alternately unequal version appears to be A122129, even A351008, odd A122130.
The alternately equal version is A351003, even A351012, odd A000009.
The alternately equal and unequal version is A351005, even A035457, odd A351593.
The alternately unequal and equal version is A351006, even A351007, odd A053251. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 4. - Peter Bala, Jan 02 2025

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^9 + q^49 + 2*q^89 + 2*q^129 + 3*q^169 + 4*q^209 + 6*q^249 + ...

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(d), p. 591.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
M. D. Hirschhorn, Some partition theorems of the Rogers-Ramanujan type, J. Combin. Theory Ser. A 27 (1979), no. 1, 33-37. MR0541341 (80j:05010). See Theorem 2. [From _N. J. A. Sloane_, Mar 19 2012]
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. A000009, A003242, A035363, A053251, A122129, A122130, A122134.

f:=n->1/mul(1-q^(20*k+n),k=0..20);
f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19);
series(%,q,200); seriestolist(%); # N. J. A. Sloane, Mar 19 2012

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, -x^5] QPochhammer[ x^4, -x^5] QPochhammer[-x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Nov 12 2016 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+1))*(1 - x^(20*k+2))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+8))*(1 - x^(20*k+9))*(1 - x^(20*k+11))*(1 - x^(20*k+12))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+18))*(1 - x^(20*k+19)) ), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1) - 1) \2, x^(k^2 + k) / prod(i=1, 2*k+1, 1 - x^i, 1 + x * O(x^(n-k^2-k)))), n))};

Expansion of f(x^2, x^8) / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 12 2016
Expansion of f(-x^3, -x^7) * f(-x^4, -x^16) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1))).
Let f(n) = 1/Product_{k >= 0} (1-q^(20k+n)). Then g.f. is f(1)*f(2)*f(5)*f(6)*f(8)*f(9)*f(11)*f(12)*f(14)*f(15)*f(18)*f(19); - N. J. A. Sloane, Mar 19 2012.
a(n) ~ (3 + sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

A045828 One fourth of theta series of cubic lattice with respect to face.

Original entry on oeis.org

1, 2, 2, 4, 3, 2, 6, 4, 4, 6, 4, 4, 7, 8, 2, 8, 8, 4, 10, 4, 4, 10, 10, 8, 9, 4, 6, 12, 8, 6, 10, 12, 4, 14, 8, 4, 16, 10, 8, 8, 9, 10, 12, 12, 8, 12, 12, 4, 20, 10, 6, 20, 8, 6, 10, 12, 8, 20, 18, 8, 11, 12, 12, 16, 8, 6, 20, 16, 12, 14, 8, 12, 20, 14, 6, 12, 20, 8, 26, 12, 8, 22, 8, 12, 15

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of solutions to n = t1 + t2 + 2*t3 where  t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - Michael Somos, Jun 29 2012

			G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 2*x^5 + 6*x^6 + 4*x^7 + 4*x^8 + 6*x^9 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + 3*q^9 + 2*q^11 + 6*q^13 + 4*q^15 + 4*q^17 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A005877, A005884, A033761, A010054, A033763, A212885.

a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* Michael Somos, Jun 29 2012 *)
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 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^4 + A)^2 / eta(x + A)^2, n))}; /* Michael Somos, Oct 25 2006 */

Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 02 2006
Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 29 2012
Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - Michael Somos, Mar 05 2003
Convolution of A033761 and A010054. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - Michael Somos, Sep 08 2018

Edited by Michael Somos, Mar 05 2003

A050470 a(n) = Sum_{d|n, n/d == 1 (mod 4)} d^2 - Sum_{d|n, n/d == 3 (mod 4)} d^2.

Original entry on oeis.org

1, 4, 8, 16, 26, 32, 48, 64, 73, 104, 120, 128, 170, 192, 208, 256, 290, 292, 360, 416, 384, 480, 528, 512, 651, 680, 656, 768, 842, 832, 960, 1024, 960, 1160, 1248, 1168, 1370, 1440, 1360, 1664, 1682, 1536, 1848, 1920, 1898, 2112, 2208, 2048, 2353, 2604

Offset: 1

N. J. A. Sloane, Dec 23 1999

Number 7 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Multiplicative because it is the Dirichlet convolution of A000290 = n^2 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

			G.f. = q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + 64*q^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A050461, A050465, A101455, A120030, A153071.
Cf. A027750, A000122, A000700, A010054, A121373.
Glaisher's E'_i (i=0..12): A002654, A050469, this sequence, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

a050470 n = a050461 n - a050465 n  -- Reinhard Zumkeller, Mar 06 2012

Basis( ModularForms( Gamma1(4), 3), 51) [2]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2]^3 (QPochhammer[ q^4] / QPochhammer[ q])^2)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^2 / 4)^2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[n/d, 2] (-1)^Quotient[n/d, 2], {d, Divisors@n}]]; (* Michael Somos, May 17 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(2*e+2) - s[p]^(e+1))/(p^2 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

{a(n) = if( n<1, 0, sumdiv( n, d, d^2 * (n/d%2) * (-1)^(n/d\2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * (eta(x^4 + A) / eta(x + A))^4, n))}; /* Michael Somos, May 17 2015 */

from math import prod
from sympy import factorint
def A050470(n): return prod((p**(e+1<<1)-(m:=(0,1,0,-1)[p&3]))//(p**2-m) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

G.f.: Sum_{n>=1} n^2*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Aug 08 2005: (Start)
Euler transform of period 4 sequence [ 4, -2, 4, -6, ...].
Expansion of eta(q^2)^6 * eta(q^4)^4 / eta(q)^4 in powers of q.
G.f.: x Product_{k>0} (1 + x^k)^4 * (1 - x^(2*k))^2 * (1 - x^(4*k))^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 8*v) * (v - 4*w) - v^2 * (v - 8*w)^2. (End)
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k) / (1 - x^k)^3. - Michael Somos, Sep 02 2005
Expansion of q * phi(q)^2 * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120030.
a(n) = A050461(n) - A050465(n). - Reinhard Zumkeller, Mar 06 2012
Multiplicative with a(p^e) = ((p^2)^(e+1) - Chi(p)^(e+1))/(p^2 - Chi(p)), Chi = A101455. - Jianing Song, Oct 30 2019
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/32 (A153071). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^2*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

A092673 a(n) = moebius(n) - moebius(n/2) where moebius(n) is zero if n is not an integer.

Original entry on oeis.org

1, -2, -1, 1, -1, 2, -1, 0, 0, 2, -1, -1, -1, 2, 1, 0, -1, 0, -1, -1, 1, 2, -1, 0, 0, 2, 0, -1, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, 1, 0, -1, -2, -1, -1, 0, 2, -1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 1, 2, -1, 1, -1, 2, 0, 0, 1, -2, -1, -1, 1, -2, -1, 0, -1, 2, 0, -1, 1, -2, -1, 0, 0, 2, -1, 1, 1, 2, 1, 0, -1, 0, 1, -1, 1, 2, 1, 0, -1, 0, 0, 0, -1, -2, -1, 0, -1, 2

Offset: 1

Jon Perry, Mar 02 2004

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Setting x=1 gives us phi(n) (A000010). Setting x=0 gives A092674.
Apparently the Dirichlet inverse of A001511. - R. J. Mathar, Dec 22 2010
Given A = A115359 as an infinite lower triangular matrix and B = the Mobius sequence as a vector, A092673 = A*B. - Gary W. Adamson, Mar 14 2011
Empirical: Letting M(n) denote the n X n matrix whereby the (i,j)-entry of M(n) is Sum_{k=1..j} floor(i/k), we have that a(n) is the (n,1)-entry of the inverse of M(n). - John M. Campbell, Aug 30 2017
John Campbell's statement is proved at the Mathematics Stack Exchange link. - Sungjin Kim, Jul 17 2019

			The first few s[n] are:
x, -2*x + 3, -x + 3, x + 1, -x + 5, 2*x, -x + 7, 4, 6, 2*x + 2, -x + 11, -x + 5, -x + 13, 2*x + 4, x + 7, 8, -x + 17, 6, -x + 19, -x + 9, x + 11, 2*x + 8, -x + 23, 8, 20, 2*x + 10, 18, -x + 13, -x + 29, -2*x + 10, -x + 31, 16, x + 19.
x - 2*x^2 - x^3 + x^4 - x^5 + 2*x^6 - x^7 + 2*x^10 - x^11 +...

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Mathematics Stack Exchange, What is the inverse of ... ?.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008683 (moebius(n)), A092149 (partial sums), A092674, A115359.

Cf. A000010, A000122, A000700, A010054, A092673, A115359, A121373.

A092673:= proc(n) if n::odd then numtheory:-mobius(n) else numtheory:-mobius(n) - numtheory:-mobius(n/2) fi end proc:
map(A092673, [$1..100]); # Robert Israel, Dec 31 2015

f[n_] := MoebiusMu[n] - If[OddQ@n, 0, MoebiusMu[n/2]]; Array[f, 105] (* Robert G. Wilson v *)

s=vector(2000); t(n)=binomial(n+1,2); s[1]=x; for(i=2,2000, s[i]=t(i)-sum(j=1,i-1, s[j]*floor(i/j))); for(i=1,2000,print1(","polcoeff(s[i],1)))

{a(n) = if( n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2)))} /* Michael Somos, Mar 26 2007 */

{a(n) = local(A, B, m); if( n<1, 0, A = x * O(x^n); B = 1 + x + A; for( k=1, n, B *= eta(x^k + A)^( m = polcoeff(B, k))); m)} /* Michael Somos, Mar 26 2007 */

a(n)=my(o=valuation(n%8,2)); if(o==0, moebius(n), if(o==1, 2*moebius(n), if(o==2, moebius(n/4), 0))) \\ Charles R Greathouse IV, Feb 07 2013

from sympy import mobius
def A092673(n): return mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

Let t(n) = binomial(n+1,2); s[1]=x; for i >= 2, s[i] = t(i)-Sum_{j=1..i-1} s[j]*floor(i/j); a(n) = coefficient of x in s[n]. - Jon Perry
a(n) is multiplicative with a(2)= -2, a(4)= 1, a(2^e)= 0 if e>2. a(p)= -1, a(p^e)= 0 if e>1, p>2. - Michael Somos, Mar 26 2007
a(8*n)= 0. a(2*n + 1) = moebius(2*n + 1). a(2*n) = moebius(2*n) - moebius(n). - Michael Somos, Mar 26 2007
|a(n)| <= 2.
1 / (1 + x) = Product_{k>0} f(-x^k)^a(k) where f() is a Ramanujan theta function. - Michael Somos, Mar 26 2007
Dirichlet g.f.: (1-2^(-s))/zeta(s). - Ralf Stephan, Mar 24 2015
G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} A(x^k). - Seiichi Manyama, Mar 31 2023
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 9/Pi^2 = 0.9118906... . - Amiram Eldar, Jan 19 2024

A122134 Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 18, 24, 28, 36, 42, 54, 62, 78, 91, 112, 130, 159, 184, 222, 258, 308, 356, 424, 488, 576, 664, 778, 894, 1044, 1196, 1389, 1590, 1838, 2098, 2419, 2754, 3162, 3596, 4114, 4668, 5328, 6032, 6864, 7760, 8806, 9936, 11252

Offset: 0

Michael Somos, Aug 21 2006, Oct 10 2007

Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
In Watson 1937 page 275 he writes "Psi_0(q^{1/2},q) = prod_1^oo (1+q^{2n}) G(-q^2)" so this is the expansion in powers of q^2. - Michael Somos, Jun 29 2015
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).
From Gus Wiseman, Feb 26 2022: (Start)
Conjecture: Also the number of even-length integer partitions y of n such that y_i != y_{i+1} for all even i. For example, the a(2) = 1 through a(9) = 7 partitions are:
  (11)  (21)  (22)  (32)  (33)    (43)    (44)    (54)
              (31)  (41)  (42)    (52)    (53)    (63)
                          (51)    (61)    (62)    (72)
                          (2211)  (3211)  (71)    (81)
                                          (3311)  (3321)
                                          (4211)  (4311)
                                                  (5211)
This appears to be the even-length version of A122135.
The odd-length version is A351595.
Cf. A035363, A035457, A350842, A350844, A351003-A351012, A351593. (End)
For Wiseman's conjecture above and three other partition-theoretic interpretations of this sequence see Connor, Proposition 3. - Peter Bala, Jan 02 2025

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = q + q^81 + q^121 + 2*q^161 + 2*q^201 + 4*q^241 + 4*q^281 + ...

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(c), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.6). MR0858826 (88b:11063)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Willard G. Connor, Partition Theorems Related to Some Identities of Rogers and Watson, Transactions of the American Mathematical Society, Vol. 214 (Dec., 1975), pp. 95-111.
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.
Mircea Merca, From a Rogers's identity to overpartitions, Periodica Mathematica Hungarica, Vol. 75, issue 2, 172-179, 2017.
Michael Somos, Introduction to Ramanujan theta functions
G. N. Watson, The Mock Theta Functions (2), Proceedings of the London Mathematical Society, s2-42: 274-304, 1937.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A053251, A122129, A122130, A122135.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2 + k) / QPochhammer[ x, x, 2 k], {k, 0, (Sqrt[ 4 n + 1] - 1) / 2}], {x, 0, n}]]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient [ 1 / (QPochhammer[ x^4, -x^5] QPochhammer[ -x, -x^5] QPochhammer[ x, x^2]), {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, -x^5] QPochhammer[ -x^3, -x^5] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Jun 29 2015 *)
nmax = 50; CoefficientList[Series[Product[1/((1 - x^(20*k+2))*(1 - x^(20*k+3))*(1 - x^(20*k+4))*(1 - x^(20*k+5))*(1 - x^(20*k+6))*(1 - x^(20*k+7))*(1 - x^(20*k+13))*(1 - x^(20*k+14))*(1 - x^(20*k+15))*(1 - x^(20*k+16))*(1 - x^(20*k+17)) *(1 - x^(20*k+18))), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2016 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n + 1) - 1)\2, x^(k^2 + k) / prod(i=1, 2*k, 1 - x^i, 1 + x * O(x^(n -k^2-k)))), n))};

Euler transform of period 20 sequence [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, ...].
Expansion of f(x^4, x^6) / f(-x^2, -x^3) in powers of x where f(, ) is the Ramanujan general theta function. - Michael Somos, Jun 29 2015
Expansion of f(-x^2, x^3) / phi(-x^2) in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Jun 29 2015
Expansion of G(-x) / chi(-x) in powers of x where chi() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jun 29 2015
G.f.: Sum_{k>=0} x^(k^2 + k) / ((1 - x) * (1 - x^2) * ... * (1 - x^(2*k))).
Expansion of f(-x, -x^9) * f(-x^8, -x^12) / ( f(-x) * f(-x^20) ) in powers of x where f(, ) is the Ramanujan general theta function.
a(n) = number of partitions of n into parts that are each either == 2, 3, ..., 7 (mod 20) or == 13, 14, ..., 18 (mod 20). - Michael Somos, Jun 29 2015 [corrected by Vaclav Kotesovec, Nov 12 2016]
a(n) ~ (3 - sqrt(5))^(1/4) * exp(Pi*sqrt(2*n/5)) / (4*sqrt(5)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2016

A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 2, 0, 2, 2, 1, 4, 0, 2, 3, 2, 2, 0, 4, 2, 2, 0, 0, 2, 1, 4, 2, 2, 2, 2, 3, 2, 0, 2, 2, 2, 2, 0, 2, 4, 4, 0, 0, 0, 1, 2, 4, 0, 2, 4, 2, 2, 1, 6, 0, 2, 2, 0, 0, 2, 4, 2, 0, 2, 2, 0, 4, 0, 4, 2, 1, 2, 0, 2, 4, 0, 0, 2, 2, 4, 3, 4, 0, 2, 2, 2, 2

Offset: 0

Michael Somos, Oct 17 2006

nonn

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

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 2*q^73 + 2*q^97 + 2*q^109 + ...

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. A093829, A131961, A131962, A248886.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)

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

Expansion of q^(-1/12) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -2, ...].
a(n) = A093829(12*n + 1).
a(n) = (-1)^n * A248886(n). a(2*n) = A131961(n). a(2*n + 1) = 2 * A131963(n). - Michael Somos, Oct 01 2015

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