A129402 -id:A129402 - cp's OEIS frontend

A000377 Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

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

Offset: 0

N. J. A. Sloane

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

Number 42 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + q + q^2 + q^3 + q^4 + 2*q^5 + q^6 + 2*q^7 + q^8 + q^9 + 2*q^10 + ...

George E. Andrews, editor, P. A. MacMahon: Collected Papers Volume II, MIT Press, 1986, p. 260.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 81, Eq. (32.5).

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, Nathan Fine 1916-1994, Notices Amer. Math. Soc., 42 (No. 6, 1995), 678-679.
Alexander Berkovich and Hamza Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, The Ramanujan Journal, Vol. 20, No. 3 (2009), pp. 375-408; arXiv preprint, arXiv:math/0611300 [math.NT], 2006-2007.
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Michael Gilleland, Some Self-Similar Integer Sequences.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Fine's Equation.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A129402, A190611.

A := Basis( ModularForms( Gamma1(24), 1), 102); A[1] + A[2] + A[3] + A[4] + A[5] + 2*A[6] + A[7] + 2*A[8] + A[9] + A[10] + 2*A[11] + 2*A[12]; /* Michael Somos, May 17 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -6, #] &]] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] + EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, May 17 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^3] QPochhammer[ q^8] QPochhammer[ -q, q] / QPochhammer[ -q^12, q^12] , {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -6, d)))};

{a(n) = if( n<1, n==0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -6, p) * X)))[n])};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^24 + A)), n))};

Expansion of (phi(q) * phi(q^6) + phi(q^2) * phi(q^3)) / 2 = psi(-q^2) * psi(-q^3) * chi(-q^6) * chi(-q^12) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jan 26 2006
Expansion of eta(q^2) * eta(q^3) * eta(q^8) * eta(q^12) / (eta(q) * eta(q^24)) in powers of q.
Multiplicative with a(0) = 1, a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24). - Michael Somos, Jun 17 2005
Moebius transform is period 24 sequence [ 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Jan 26 2006
Euler transform of period 24 sequence [ 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 22 2011
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(8*k)) / (1 + x^(12*k)).
G.f.: 1 + Sum_{k>0} x^k * (1 + x^(4*k)) * (1 + x^(6*k)) / (1 + x^(12*k)). - Michael Somos, Sep 10 2005
G.f.: 1 + Sum{n = -infinity...infinity} (q^n + q^(5*n)) / (1 + q^(12*n)) (see Berkovich/Yesilyurt). - Ralf Stephan, May 14 2007
a(n) = (-1)^n * A190611(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0. a(2*n) = a(3*n) = a(n). a(2*n + 1) = A129402(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.2825... . - Amiram Eldar, Oct 23 2022

Edited by Michael Somos, Sep 10 2002

A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, -2, 1, -2, 0, -2, 0, 0, 2, 0, 3, -1, 2, -2, 2, -4, 0, 0, 0, 0, 2, 0, 3, 0, 2, -4, 0, -2, 0, -2, 0, 0, 0, 0, 2, -3, 4, -2, 1, -2, 0, -2, 0, 0, 2, 0, 2, -2, 2, -2, 4, -2, 0, 0, 0, 0, 0, 0, 3, 0, 4, -2, 0, -2, 0, -2, 0, 0, 0, 0, 4, -3, 2, -2, 0, -4, 0

Offset: 0

Michael Somos, May 14 2011

sign

Number 63 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).

			G.f. = 1 - x + 2*x^2 - 2*x^3 + x^4 - 2*x^5 - 2*x^7 + 2*x^10 + 3*x^12 - x^13 + ...
G.f. = q - q^3 + 2*q^5 - 2*q^7 + q^9 - 2*q^11 - 2*q^15 + 2*q^21 + 3*q^25 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
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
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113700, A128580, A128583, A128591, A129402, A134177, A257920, A259895, A259896, A260089, A260110, A260118, A260308.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ -x^3] / (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^12]), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, kronecker( -6, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^4 * eta(x^24 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^3), n))};

{a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, (-1)^e, p%24 < 12, (e+1) * if( p%24 < 6, 1, (-1)^e), (1 + (-1)^e) / 2 )))};

Expansion of phi(-x^3) * psi(x^4) - x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q) * eta(q^4)^4 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3) * eta(q^8) * eta(q^12)^3) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, 2, 0, -2, -1, -1, -1, -1, 0, 2, -1, -2, -1, 2, 0, -1, -1, -1, -1, -2, 0, 2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker( 6, k) * q^k / (1 + q^(2*k)) = Sum_{k>=0} a(k) * q^(2*k + 1).
G.f.: Product_{k>0}  (1 + (-x)^k) * (1 - (-x^2)^k) * (1 - (-x^3)^k) * (1 + (-x^6)^k).
a(n) = (-1)^n * A129402(n). a(3*n + 1) = -a(n). a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
a(12*n) = A113700(n). a(12*n + 2) = 2 * A128583(n). a(12*n + 5) = -2 * A128591(n). - Michael Somos, Jun 09 2015
a(n) = (-1)^floor(n/2) * A128580(n) = (-1)^(n + floor(n/2)) * A134177(n). - Michael Somos, Jul 29 2015
a(3*n) = A260110(n). a(3*n + 2) = 2 * A260118(n). - Michael Somos, Jul 29 2015
a(4*n) = A260308(n). a(4*n + 1) = - A257920(n). a(4*n + 2) = 2 * A259895(n). a(4*n + 3) = -2 * A259896(n). - Michael Somos, Jul 29 2015
a(12*n + 3) = -2 * A260089(n). - Michael Somos, Jul 29 2015

A113780 Number of solutions to 24n+1 = x^2+24y^2, x a positive integer, y an integer.

Original entry on oeis.org

1, 3, 3, 2, 2, 3, 4, 1, 2, 4, 2, 4, 1, 2, 2, 1, 8, 2, 2, 2, 0, 4, 1, 4, 2, 2, 5, 4, 2, 0, 4, 4, 2, 0, 0, 3, 4, 4, 4, 2, 3, 4, 2, 2, 4, 0, 0, 2, 2, 4, 2, 9, 2, 0, 2, 2, 4, 1, 4, 0, 4, 4, 2, 0, 4, 4, 4, 2, 0, 2, 1, 8, 0, 2, 2, 2, 6, 1, 2, 4, 0, 4, 4, 2, 2, 0, 8, 2, 2, 2, 2, 0, 1, 8, 0, 2, 4, 0, 0, 2, 5, 6, 4, 2, 4

Offset: 0

Christian G. Bower, Jan 20 2006, based on a message from Dean Hickerson

nonn

If 24*n+1 is not a square or if sqrt(24*n+1) == 1 or 11 (mod 12), then A000009(n) == a(n) (mod 4), otherwise A000009(n) == a(n) + 2 (mod 4).
Implied by the arithmetic of Q[sqrt(-6)]: Let 24*n+1 = p_1^e_1 * ... * p_r^e_r * q_1^f_1 * ... * q_s^f_s, where the p_i's are distinct primes == 1, 5, 7, or 11 (mod 24) and the q_i's are distinct primes == 13, 17, 19, or 23 (mod 24). If some f_i is odd, then a(n) = 0. Otherwise, a(n) = (e_1 + 1) * ... * (e_r + 1). a(n) == 2 (mod 4) iff all of the f_i's are even and all but one of the e_i's are even and the one e_i which is odd is == 1 (mod 4). Since A000009(n) and a(n) are both odd if 24*n+1 is a square, we can replace a by A000009 in this.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			If n=51, the solutions (x,y) are: (7,+-7), (19,+-6), (25,+-5), (29,+-4), (35,0) so a(51)=9.
G.f. = 1 + 3*x + 3*x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + x^7 + 2*x^8 + 4*x^9 + ...
G.f. = q + 3*q^25 + 3*q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + q^169 + 2*q^193 + ...

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. A001318 generalized pentagonal numbers, indices of odd values of a(n) and A000009.
Cf. A114913 = values k such that A000009(k) == 2 (mod 4) and such that a(k) == 2 (mod 4).
Cf. A128580, A129402, A134177, A190615.

a[ n_] := If[ n < 0, 0, With[{m = 24 n + 1}, Sum[ KroneckerSymbol[ -12, d] KroneckerSymbol[ 2, m/d], {d, Divisors @ m}]]]; (* Michael Somos, Jun 08 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 08 2013 *)

{a(n) = if( n<0, 0, n = 24*n + 1; sumdiv( n, d, kronecker( -12, d) * kronecker( 2, n/d)))}; /* Michael Somos, Mar 11 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Jun 08 2012 */

Expansion of phi(x) * phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Jun 08 2012
Expansion of f(x, x) * f(x, x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jun 08 2013
Expansion of eta(q^2)^6 * eta(q^3)^2 / (eta(q)^3 * eta(q^4)^2 * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 12 sequence [ 3, -3, 1, -1, 3, -4, 3, -1, 1, -3, 3, -2, ...]. - Michael Somos, Jun 08 2012
a(n) = A128580(12*n) = A129402(12*n) = A134177(12*n) = A190615(12*n). - Michael Somos, Jun 08 2012

A190611 Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 1, -1, 1, -2, 1, -2, 1, -1, 2, -2, 1, 0, 2, -2, 1, 0, 1, 0, 2, -2, 2, 0, 1, -3, 0, -1, 2, -2, 2, -2, 1, -2, 0, -4, 1, 0, 0, 0, 2, 0, 2, 0, 2, -2, 0, 0, 1, -3, 3, 0, 0, -2, 1, -4, 2, 0, 2, -2, 2, 0, 2, -2, 1, 0, 2, 0, 0, 0, 4, 0, 1, -2, 0, -3, 0, -4, 0

Offset: 0

Michael Somos, May 14 2011

sign

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

			G.f. = 1 - q + q^2 - q^3 + q^4 - 2*q^5 + q^6 - 2*q^7 + q^8 - q^9 + 2*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000377, A129402.

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n DivisorSum[ n, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q] QPochhammer[ -q^3] QPochhammer[ q^8] QPochhammer[ q^12, -q^12], {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<1, n==0, (-1)^n * sumdiv( n, d, kronecker( -6, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3 * eta(x^8 + A) / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^24 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^3 * eta(q^8) / (eta(q^2)^2 * eta(q^3) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ -1, 1, 0, 0, -1, -1, -1, -1, 0, 1, -1, -2, -1, 1, 0, -1, -1, -1, -1, 0, 0, 1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 96^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129402.
a(n) = (-1)^n * A000377(n). a(24*n + 13) = a(24*n + 17) = a(24*n + 19) = a(24*n + 23) = 0.
a(2*n) = A000377(n). a(2*n + 1) = - A129402(n). a(3*n) = a(n). - Michael Somos, Nov 11 2015

A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 12 2015

nonn

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

			G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...

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. A000377, A128591, A129402, A134177, A192013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

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

Expansion of q^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).
a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).

A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 07 2015

nonn

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

			G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

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. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)

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

{a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015

A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 0, 3, 2, 0, 3, 0, 0, 2, 1, 0, 2, 4, 0, 3, 0, 0, 4, 0, 0, 1, 2, 0, 2, 0, 0, 4, 3, 0, 2, 2, 0, 4, 0, 0, 1, 2, 0, 2, 2, 0, 2, 0, 0, 1, 0, 0, 8, 2, 0, 2, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 0, 4, 0, 0, 1, 2, 0, 4, 0, 0, 2, 0, 0, 2, 4, 0, 5, 0, 0, 4, 2, 0, 2, 2, 0

Offset: 0

Michael Somos, Jul 22 2015

nonn

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

			G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...
G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...

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. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.

a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};

{a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};

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

Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].
a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).

A261115 Expansion of f(x, x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 3, 2, 0, 0, 3, 4, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 2, 0, 0, 4, 2, 0, 0, 1, 6, 0, 0, 2, 2, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 1, 4, 0, 0, 2, 4, 0, 0, 2, 4, 0, 0, 1, 2, 0, 0, 8, 0, 0, 0, 2, 4, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 4, 4, 0

Offset: 0

Michael Somos, Aug 08 2015

nonn

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

			G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 3*x^8 + 4*x^9 + 2*x^12 + 2*x^13 + 2*x^16 + ...
G.f. = q + 2*q^7 + 3*q^25 + 2*q^31 + 3*q^49 + 4*q^55 + 2*q^73 + 2*q^79 + ...

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. A113780, A115660, A128580, A128581, A129402, A192013, A260110.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^24 + A)), n))};

Expansion of q^(-1/6) * eta(q^2)^5 * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -2, ...].
a(n) = (-1)^n * A260110(n) = A128580(3*n) = A129402(3*n) = A115660(6*n + 1) = A128581(6*n + 1) = A192013(6*n + 1).
a(4*n) = A113780(n). a(4*n + 1) = 2 * A260089(n). a(4*n + 2) = a(4*n + 3) = 0.

A261118 Expansion of psi(x)^2 * psi(-x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 08 2015

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^5 + 3*x^6 + 2*x^7 + 2*x^8 + 2*x^11 + 3*x^12 + ...
G.f. = q + 2*q^5 + q^9 + 2*q^21 + 3*q^25 + 2*q^29 + 2*q^33 + 2*q^45 + ...

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. A129402, A190615, A192013, A259668, A259895, A260308.

a[n_]:= SeriesCoefficient[(-1)^(-1/8)*q^(-1/4)*(EllipticTheta[2, 0, Sqrt[q]]*EllipticTheta[2, 0, I*Sqrt[q^3]])^2/(8*EllipticTheta[3, 0, -q^4]*EllipticTheta[2, 0, I*q^3]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^2 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^24 + A)), n))};

Expansion of f(-x^8) * f(x, x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^4 * eta(q^3)^2 * eta(q^8) * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 0, 0, 2, -1, 2, -1, 0, -2, 2, -2, 2, -2, 0, -1, 2, -1, 2, 0, 0, -2, 2, -2, ...].
a(n) = (-1)^n * A259668(n) = A129402(2*n) = A190615(2*n) = A192013(4*n) = A000377(4*n + 1) = A129402(6*n + 1).
a(2*n) = A260308(n). a(2*n + 1) = 2 * A259895(n).

A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 08 2015

nonn

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

			G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ...
G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ...

G. C. Greubel, 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. A000377, A129402, A192013, A257920, A257921, A259896, A261118.

a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Dec 22 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))};

a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ Michel Marcus, Dec 13 2017

Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function.
Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...].
a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3).
a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n).

cp's OEIS Frontend

A000377 Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

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A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.

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A113780 Number of solutions to 24*n+1 = x^2+24*y^2, x a positive integer, y an integer.

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A190611 Expansion of f(q^3) * f(-q^8) * chi(-q^12) / chi(q) in powers of q where f(), chi() are Ramanujan theta functions.

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A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

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A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

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A113780 Number of solutions to 24n+1 = x^2+24y^2, x a positive integer, y an integer.