A112605 -id:A112605 - cp's OEIS frontend

A164273 Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 2, -2, 0, 0, 4, 0, -2, 0, 0, -2, -4, 0, 0, 6, 0, 0, 4, 0, -4, 0, 0, 0, -2, 0, 2, -4, 0, 0, 4, 0, 0, 0, 0, -2, -4, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, -6, 0, 0, -4, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 2, -4, 0, 0, 4, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 8, 0, -4, 0, 0, 0, -4, 0, 0, -2, 0, 0, 4, 0

Offset: 0

Michael Somos, Aug 11 2009

sign

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

			G.f. = 1 - 2*q + 2*q^3 - 2*q^4 + 4*q^7 - 2*q^9 - 2*q^12 - 4*q^13 + 6*q^16 + ...

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. A164272.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0 ,q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164273[n_] := SeriesCoefficient[f[-q, -q]*f[q^3, q^3], {q, 0, n}]; Table[A164273[n], {n, 0, 50}] (* G. C. Greubel, Sep 16 2017 *)

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

Expansion of eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ -2, -1, 0, -1, -2, -4, -2, -1, 0, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112605.
a(n) = (-1)^n * A164272(n).

A253243 Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -2, 2, 0, -4, 1, 0, 0, 2, 0, 0, 3, 0, -4, 2, 0, 0, 2, 0, -2, 0, 0, -4, 2, 0, 0, 2, 0, -4, 1, 0, -4, 4, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, -4, 2, 0, -4, 0, 0, 0, 4, 0, -2, 2, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0, -8, 2, 0, 0, 1, 0, 0, 4, 0, -4, 2, 0, 0, 2

Offset: 0

Michael Somos, Jun 04 2015

sign

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

			G.f. = 1 - 2*x^2 + 2*x^3 - 4*x^5 + x^6 + 2*x^9 + 3*x^12 - 4*x^14 + 2*x^15 + ...
G.f. = q - 2*q^9 + 2*q^13 - 4*q^21 + q^25 + 2*q^37 + 3*q^49 - 4*q^57 + ...

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. A005928, A097109, A112605, A112848, A123477, A123530, A123884, A226535, A246650.

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

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, [ 0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9 + 1]))};

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

Expansion q^(-1/4) * eta(q^2)^2 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, -2, 2, -1, 0, -4, 0, -1, 2, -2, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246650.
a(n) = A123530(2*n) = A097109(4*n + 1) = A112848(4*n + 1) = A123477(4*n + 1). 3 * a(n) = A226535(4*n + 1). -3 * a(n) = A005928(4*n + 1).
a(3*n) = A123884(n). a(3*n + 1) = 0. a(3*n + 2) = -2 * A112605(n).

A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 16 2007

sign

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

Number 53 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - x - 2*x^3 + x^4 + 2*x^6 - 2*x^9 + 2*x^10 + x^12 - x^13 - 2*x^15 + ...
G.f. = q - q^3 - 2*q^7 + q^9 + 2*q^13 - 2*q^19 + 2*q^21 + q^25 - q^27 + ...

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. A033762, A112604, A113605, A112606, A112608, A112609, A121361, A123884, A129451.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := With[ {m = 2 n + 1}, If[ m < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -4, m/d], {d, Divisors[ m]}]]]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, d) * kronecker( 12, n/d)))};

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

Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 0, -1, -1, -2, 0, -1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 11 (mod 12), b(p^e) = e+1 if p == 1 (mod 12), b(p^e) = (-1)^e * (e+1) if p == 7 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 48^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A033762(n). a(2*n) = A112604(n). a(2*n + 1) = -A112605(n). a(3*n) = A129451(n). a(3*n + 1) = -a(n). a(3*n + 2) = 0.
a(4*n) = A112606(n). a(4*n + 1) = - A112608(n). a(4*n + 2) = 2 * A112607(n). a(4*n + 3) = - 2 * A112609(n).
a(6*n) = A123884(n). a(6*n + 3) = -2 * A121361(n).

A136748 Expansion of (a(q) - a(q^2) - 4a(q^4) + 4a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 1, -3, 0, -1, 2, 3, 1, 0, 0, -3, 2, -2, 0, -3, 0, -1, 2, 0, 2, 0, 0, 3, 1, -2, 1, -6, 0, 0, 2, 3, 0, 0, 0, -3, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 0, -3, 3, -1, 0, -6, 0, -1, 0, 6, 2, 0, 0, 0, 2, -2, 2, -3, 0, 0, 2, 0, 0, 0, 0, 3, 2, -2, 1, -6, 0, -2, 2, 0

Offset: 1

Michael Somos, Jan 22 2008

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - q^2 + q^3 - 3*q^4 - q^6 + 2*q^7 + 3*q^8 + q^9 - 3*q^12 + 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033762, A093829, A097195, A112604, A112605, A123484.

Cf. A004016, A005882, A005928.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (Mod[#, 2] - 4 Boole[Mod[#, 8] == 4]) KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 12 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1 || # == 3, 1, # == 2, If[#2 < 2, -1, -3 (-1)^#2], Mod[#, 6] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* Michael Somos, Oct 12 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3+A) * eta(x^4 + A)^4 * eta(x^24 + A)^2 / ( eta(x^2 + A) * eta(x^8+A) * eta(x^12+A) )^2, n))};

{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==2, if( e<2, -1, -3 * (-1)^e), p==3, 1, p%6>1, !(e%2), e+1)))};

{a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) -4 * (d%8==4)) * kronecker(-3, n/d)))};

Expansion of eta(q) * eta(q^3) * eta(q^4)^4 * eta(q^24)^2 / (eta(q^2) * eta(q^8) * eta(q^12))^2 in powers of q.
Euler transform of period 24 sequence [ -1, 1, -2, -3, -1, 0, -1, -1, -2, 1, -1, -2, -1, 1, -2, -1, -1, 0, -1, -3, -2, 1, -1, -2, ...].
a(n) is multiplicative with a(2) = -1, a(2^e) = -3 * (-1)^e if e>1, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123484.
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k)) * (1 - x^(4*k) + x^(8*k))^2.
Moebius transform is period 24 sequence [ 1, -2, 0, -2, -1, 0, 1, 6, 2, -1, 0, 1, -2, 0, -6, -1, 0, 1, 2, 0, 2, -1, 0, ...].
a(2*n) = A244375(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n).
a(4*n) = -3 * A093829(n). a(4*n + 1) = A112604(n). a(4*n + 2) = -A033762(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
Expansion of q * f(-q, -q) * f(q^2, q^10) / f(-q, -q^5)^2 in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 12 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi*sqrt(3)/4) * n. - Amiram Eldar, Jan 28 2024

A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 29 2008

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...

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. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.

Cf. A035178, A093829, A113447 are same up to sign.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A))) / 3, n))}; /* Michael Somos, May 06 2015 */

Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless e=0, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = -(-1)^n * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n).

A164272 Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, -2, -2, 0, 0, -4, 0, 2, 0, 0, -2, 4, 0, 0, 6, 0, 0, -4, 0, 4, 0, 0, 0, 2, 0, -2, -4, 0, 0, -4, 0, 0, 0, 0, -2, 4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, 6, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, 4, 0, -2, -4, 0, 0, -4, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, -8, 0, 4, 0, 0, 0, 4, 0, 0, -2, 0, 0, -4, 0

Offset: 0

Michael Somos, Aug 11 2009

sign

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

			G.f. = 1 + 2*q - 2*q^3 - 2*q^4 - 4*q^7 + 2*q^9 - 2*q^12 + 4*q^13 + 6*q^16 + ...

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. A112604, A112605, A115978, A129449, A164273, A246752.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164272[n_] := SeriesCoefficient[f[q, q]*f[-q^3, -q^3], {q, 0, n}]; Table[A164272[n], {n,0,50}] (* G. C. Greubel, Sep 16 2017 *)

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

Expansion of eta(q^2)^5 * eta(q^3)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112604.
a(n) = (-1)^n * A164273(n).
a(2*n + 1) = 2 * A129449(n). a(4*n) = A115978(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 2) = 0. a(4*n + 3) = -2 * A112605(n).
a(3*n) = A164273(n). a(3*n + 1) = 2 * A246752(n). a(3*n + 2) = 0. - Michael Somos, Sep 02 2015

A180318 Expansion of a(-q) in powers of q where a(q) is a cubic AGM function.

Original entry on oeis.org

1, -6, 0, -6, 6, 0, 0, -12, 0, -6, 0, 0, 6, -12, 0, 0, 6, 0, 0, -12, 0, -12, 0, 0, 0, -6, 0, -6, 12, 0, 0, -12, 0, 0, 0, 0, 6, -12, 0, -12, 0, 0, 0, -12, 0, 0, 0, 0, 6, -18, 0, 0, 12, 0, 0, 0, 0, -12, 0, 0, 0, -12, 0, -12, 6, 0, 0, -12, 0, 0, 0, 0, 0, -12, 0

Offset: 0

Michael Somos, Aug 27 2010

sign

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

			G.f. = 1 - 6*q - 6*q^3 + 6*q^4 - 12*q^7 - 6*q^9 + 6*q^12 - 12*q^13 + 6*q^16 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004016, A033762, A112604. A112605.

A := Basis( ModularForms( Gamma1(12), 1), 75); A[1] - 6*A[2] - 6*A[4] + 6*A[5]; /* Michael Somos, Sep 14 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 6 Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ n]}]]; (* Michael Somos, Sep 14 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^3 - 9 q QPochhammer[ -q^9]^3) / QPochhammer[ -q^3], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)

{a(n) = if( n<1, n==0, 6 * (-1)^n * sumdiv(n, d, kronecker(d, 3)))};

Expansion of 2 * a(q^4) - a(q) in powers of q where a() is a cubic AGM theta function.
Expansion of phi(-q) * phi(-q^3) - 4 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Sep 14 2015
Expansion of theta_3(-q) * theta_3(-q^3) - theta_2(q) * theta_2(q^3) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = - (12)^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A004016(n).
G.f.: 1 + 6 * Sum_{k>0} (-x)^k/(1 + (-x)^k + x^(2*k)) = Sum_{j, k in Z} (-x)^(j*j + j*k + k*k).
a(2*n) = -6 * A033762(n). a(4*n) = A004016(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 2) = 0. a(4*n + 3) = -6 * A112605(n). - Michael Somos, Sep 14 2015

A253625 Expansion of psi(q^2) * f(-q, q^2)^2 / f(-q, -q^5) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 3, -1, 3, 0, 3, -2, 3, -1, 0, 0, 3, -2, 6, 0, 3, 0, 3, -2, 0, -2, 0, 0, 3, -1, 6, -1, 6, 0, 0, -2, 3, 0, 0, 0, 3, -2, 6, -2, 0, 0, 6, -2, 0, 0, 0, 0, 3, -3, 3, 0, 6, 0, 3, 0, 6, -2, 0, 0, 0, -2, 6, -2, 3, 0, 0, -2, 0, 0, 0, 0, 3, -2, 6, -1, 6, 0, 6, -2

Offset: 0

Michael Somos, Jan 06 2015

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - q + 3*q^2 - q^3 + 3*q^4 + 3*q^6 - 2*q^7 + 3*q^8 - q^9 + 3*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A033687, A033762, A093766, A097195, A107760, A112604, A112605, A121361, A121373, A122861, A123884, A253623, A253626.

A := Basis( ModularForms( Gamma1(12), 1), 81); A[1] - A[2] + 3*A[3] - A[4] + 3*A[5];

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^ n Sum[(-1)^ Quotient[ d, 3] {1, 1, 0}[[ Mod[d, 3, 1] ]] {1, 2}[[ Mod[n/d, 2, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ -q^3, q^6] QPochhammer[ -q^2, q^2])^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^2 EllipticTheta[ 2, 0, q]^2 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}];

{a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, (-1)^(d\3) * (d%3>0) * (2-(n\d)%2)))};

{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)^8 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^4), n))};

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

Expansion of psi(q^2)^2 * phi(q^3)^2 / (psi(q) * psi(q^3)) = f(-q) * f(-q^3) * (chi(q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (-a(q) - 3*a(q^2) + 4*a(q^4)) / 6 = b(q^4) * (b(q) + 2*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q) * eta(q^4)^4 * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -1, 3, 2, -1, -1, -2, -1, -1, 2, 3, -1, -2, ...].
Moebius transform is period 12 sequence [ -1, 4, 0, 0, 1, 0, -1, 0, 0, -4, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253623.
a(n) = -b(n) where b() is multiplicative with b(2^e) = -3 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*4) * (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 + x^(3*k))^4 / (1 - x^(2*k) + x^(4*k))^4.
a(n) = (-1)^n * A253626(n). a(2*n) = A107760(n). a(2*n + 1) = - A033762(n). a(3*n) = a(n). a(3*n + 1) = - A122861(n). a(4*n + 1) = - A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = - A112605(n).
a(6*n + 1) = - A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = - A123884(n). a(12*n + 7) = -2 * A121361(n). a(12*n + 10) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jun 08 2025

Typo in formula fixed by Colin Barker, Jan 08 2015

A227354 Expansion of 2 * a(q) - a(q^2) in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, 12, -6, 12, 12, 0, -6, 24, -6, 12, 0, 0, 12, 24, -12, 0, 12, 0, -6, 24, 0, 24, 0, 0, -6, 12, -12, 12, 24, 0, 0, 24, -6, 0, 0, 0, 12, 24, -12, 24, 0, 0, -12, 24, 0, 0, 0, 0, 12, 36, -6, 0, 24, 0, -6, 0, -12, 24, 0, 0, 0, 24, -12, 24, 12, 0, 0, 24, 0, 0, 0

Offset: 0

Michael Somos, Jul 08 2013

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 12*q - 6*q^2 + 12*q^3 + 12*q^4 - 6*q^6 + 24*q^7 - 6*q^8 + 12*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033762, A112604, A112605, A122859, A304656.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] + 3 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 6 Sum[ JacobiSymbol[ d, 3] (Mod[ n/d, 2] + 1), {d, Divisors@n}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, n==0, 12 * sumdiv( n, d, kronecker( d, 3)) - 6 * sumdiv( 2*n, d, kronecker( d, 3)))};

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 12 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, (1 + 3*(-1)^e) / 4, if( p == 3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2 ))))))};

Expansion of (4 * b(q^4)^2 - 2 * b(q) * b(q^4) - b(q)^2) / b(q^2) in powers of q where b() is a cubic AGM theta function.
Expansion of phi(-q^2)^3 / phi(-q^6) + 12 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jan 09 2015
Expansion of theta_4(q^2)^3 / theta_4(q^6) + 3 * theta_2(q) * theta_2(q^3) in powers of q.
Moebius transform is period 6 sequence [ 12, -18, 0, 18, -12, 0, ...].
a(n) = 12 * b(n) where b(n) is multiplicative with b(2^e) = (1 + 3*(-1)^e) / 4, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(n) = A122859(8*n). a(2*n) = A122859(n). a(2*n + 1) = 12 * A033762(n). a(4*n) = a(n). a(4*n + 1) = 12 * A112604(n). a(4*n + 2) = -6 * A033762(n). a(4*n + 3) = 12 * A112605(n).
G.f.: 1 + 6 * Sum_{k>0} ((k mod 2) + 1) * x^k / (1 + x^k + x^(2*k)). - Michael Somos, Jan 09 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - Amiram Eldar, Nov 23 2023

A253626 Expansion of psi(q^2) * f(q, q^2)^2 / f(q, q^5) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 06 2015

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A093602, A097195, A107760, A112604, A112605, A121361, A122861, A123884, A253625.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 86); A[1] + A[2] + 3*A[3] + A[4] + 3*A[5];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ (-1)^ Quotient[ d, 3] {1, 1, 0}[[ Mod[d, 3, 1] ]] {1, 2}[[ Mod[n/d, 2, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q^3] (QPochhammer[ q^3, q^6] QPochhammer[ -q^2, q^2])^4, {q, 0, n}];

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

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

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

Expansion of psi(q^2)^2 * phi(-q^3)^2 / (psi(-q) * psi(-q^3)) = f(q) * f(q^3) * (chi(-q^3) / chi(-q^2))^4 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (a(q) + 3*a(q^2) + 2*a(q^4)) / 6 = b(q^4) * (-b(q) + 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^3)^3 * eta(q^4)^3 / ( eta(q) * eta(q^2) * eta(q^6) * eta(q^12) ) in powers of q.
Euler transform of period 12 sequence [ 1, 2, -2, -1, 1, 0, 1, -1, -2, 2, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, 0, -1, 0, 1, 0, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) is multiplicative with a(0) = 1, a(2^e) = 3 if e > 0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (3 - (k mod 2)*2) * (q^k + q^(3*k)) / (1 + q^(2*k) + q^(4*k)).
G.f.: Product_{k>0} (1 - q^(3*k))^3 * (1 - q^(4*k))^3 / ( (1 - q^k) * (1 - q^(2*k)) * (1 - q^(6*k)) * (1 - q^(12*k)) ).
a(n) = (-1)^n * A253625(n). a(2*n) = A107760(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(4*n + 1) = A112604(n). a(4*n + 2) = 3 * A033762(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). a(12*n + 10) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Jan 21 2024

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A164273 Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.

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A253243 Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A129449 Expansion of psi(-x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

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A136748 Expansion of (a(q) - a(q^2) - 4*a(q^4) + 4*a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.

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A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

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A164272 Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A180318 Expansion of a(-q) in powers of q where a(q) is a cubic AGM function.

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A136748 Expansion of (a(q) - a(q^2) - 4a(q^4) + 4a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.