A128580 -id:A128580 - cp's OEIS frontend

A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

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: 1

Michael Somos, Jan 28 2006

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

Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012

			G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
Y. 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. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
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, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* Michael Somos, Oct 22 2015 *)

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 2, d) * kronecker( -3, n/d)))};

Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
a(n) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
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).
G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
abs(a(n)) = A000377(n). a(n) = (-1)^n * A128581(n). a(2*n) = a(3*n) = -a(n). a(2*n + 1) = A128580(n). - Michael Somos, Mar 14 2012
abs(a(n)) = A192013(n) unless n=0. - Michael Somos, Oct 22 2015
a(3*n + 1) = A263571(n). a(4*n) = A259668(n). a(6*n + 1) = A261115(n). a(6*n + 4) = A263548(n). a(8*n + 1) = A260308(n). - Michael Somos, Oct 22 2015
a(n) = A000377(n) - A108563(n) = A046113(n) - A000377(n). - Michael Somos, Oct 22 2015

A129402 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.

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

Offset: 0

Michael Somos, Apr 13 2007

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 + 2*x^14 + ...
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 + q^27 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000377, A128580, A128582, A113780, A190615, A257920, A259895, A260308, A261115, A261118, A261119, A261122.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ x^3] QPochhammer[ -x, x] QPochhammer[ x^6, -x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

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

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

Expansion of f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A190611.
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).
a(n) = A000377(2*n + 1). a(3*n + 2) = 2 * A128582(n). a(12*n) = A113780(n).
a(n) = (-1)^n * A190615(n) = (-1)^floor( (n+1) / 2) * A128580(n). - Michael Somos, Nov 11 2015
a(2*n) = A261118(n). a(2*n + 1) = A261119(n). a(3*n) = A261115(n). - Michael Somos, Nov 11 2015
a(4*n) = A260308(n). a(4*n + 1) = A257920(n). a(4*n + 2) = 2 * A259895(n). - Michael Somos, Nov 11 2015
a(n) = - A261122(4*n + 2). - Michael Somos, Nov 11 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A128582 Expansion of f(x^4, x^12) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 11 2007

nonn

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

			G.f. = 1 + x + x^4 + 2*x^5 + x^8 + x^9 + 2*x^12 + x^13 + x^16 + x^17 + 2*x^20 + ...
G.f. = q^5 + q^11 + q^29 + 2*q^35 + q^53 + q^59 + 2*q^77 + q^83 + q^101 + q^107 + ...

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. A128580, A128583, A128591.

a[ n_] := If[ n < 0, 0, With[{m = 6 n + 5}, -1/2 DivisorSum[ m, KroneckerSymbol[ -12, #] KroneckerSymbol[ 2, m/#] &]]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-7/8) EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, Pi/4, x]^2 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *)

{a(n) = if( n<0, 0, n = 6*n + 5; -1/2 * sumdiv( n, d, kronecker( -12, d) * kronecker( 2, n/d)))};

{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^8 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Expansion of psi(x) * psi(-x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q^2)^2 * eta(q^3) * eta(q^8)^2 * eta(q^12) / (eta(q) * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 0, 1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 1, 0, -1, 1, -2, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(4*k))^2 * (1 - x^(6*k - 3)) * (1 - x^(12*k)).
A128580(3*n + 2) = -2 * a(n). a(4*n) = A128583. a(4*n + 1) = A128591(n). a(4*n + 2) = a(4*n + 3) = 0.

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

A134177 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.

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, Oct 11 2007

sign

Number 64 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 + q^27 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10000
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. A113780, A128580.

a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Jun 24 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 24 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( 12, d) * kronecker( -2, n/d)))};

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

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); 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)))};

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 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).
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} Kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
a(n) = (-1)^n * A128580(n). a(12*n) = A113780(n).

A128581 Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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: 1

Michael Somos, Mar 11 2007

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

			G.f. = 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 = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A115660, A128580.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ #, 8] KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^6])/2, {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q^24] QPochhammer[ q^4, q^8] QPochhammer[ q^3, -q^3], {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)

{a(n) = -(-1)^n * sumdiv(n, d, kronecker(d, 8) * kronecker(n/d, 3))}

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

A128581(n)={prod(i=1,matsize(n=factor(n))[1], if(12>n[i,1]%24, if(bittest(12,n[i,1]),(-1)^(n[i,1]+n[i,2]-1), if(bittest(n[i,2],0)&&n[i,1]%6>1,-n[i,2]-1,n[i,2]+1)), !bittest(n[i,2],0)))} \\ a(p^e) as given in formula. - M. F. Hasler, May 07 2018

Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^24) / (eta(q) * eta(q^6)^2 * eta(q^8)) in powers of q.
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, ...].
Multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
a(n) = -(-1)^n * A115660(n). a(2*n) = A115660(n). a(2*n + 1) = A128580(n).
abs(a(n)) = A000377(n) = A192013(n). - M. F. Hasler, May 07 2018

A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 11 2007

nonn

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

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 3*x^10 + ...
G.f. = q^5 + q^29 + q^53 + 2*q^77 + q^101 + 2*q^125 + q^149 + q^173 + ...

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. A128580, A128582, A190615, A229723, A259895, A260118.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] QPochhammer[ x^6]^2 / (QPochhammer[ x] QPochhammer[ x^12] ), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 4, 0, x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];

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

Expansion of q^(-5/24) * eta(q^2) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 6^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229723.
a(n) = A128582(4*n) = A259895(3*n) = A260118(4*n). 2 * a(n) = A190615(12*n + 2). - Michael Somos, Nov 15 2015
-2 * a(n) = A128580(12*n + 2). - Michael Somos, Dec 22 2016

A128591 Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 11 2007

nonn

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

			G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + x^10 + 3*x^11 + ...
G.f. = q^11 + 2*q^35 + q^59 + q^83 + q^107 + q^131 + 2*q^155 + q^179 + 2*q^203 + ...

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. A128580, A128582, A190615, A257920, A257921,A260118.

a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *)

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

Expansion of chi(x) * psi(x) * psi(-x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Nov 15 2015
Expansion of  q^(-11/24) * eta(q^2)^4 * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 1, -1, 2, -2, 2, -1, 1, -2, 2, -2, ...].
a(n) = A128582(4*n + 1).
2 * a(n) = A257920(3*n + 1). - a(n) = A260118(4*n + 1). 2 * a(n) = A257921(6*n + 2). -2 * a(n) = A128580(12*n + 5) = A190615(12*n + 5). - Michael Somos, Nov 15 2015

A259668 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

Offset: 0

Michael Somos, Jul 02 2015

sign

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. A113780, A115660, A128580, A128581, A134177.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[x^12, x^24] QPochhammer[ x^6] (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6])^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, x^(3/2)] EllipticTheta[ 2, Pi/4, x^(1/2)])^2 / (2^(5/2) x^(1/4) EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * 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.
Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -1, -2, -1, 0, 0, -2, -2, -2, 0, 0, -1, -2, -1, -2, 0, 0, 0, -2, -2, ...].
a(n) = A128580(2*n) = A134177(2*n) = A115660(4*n) = A128581(4*n).
a(6*n + 1) = -2 * A113780(n).

cp's OEIS Frontend

A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A129402 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.

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A128582 Expansion of f(x^4, x^12) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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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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A134177 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.

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