A121361 -id:A121361 - cp's OEIS frontend

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

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).

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).

A246650 Expansion of phi(x) * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 31 2014

sign

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^4 - 3*x^5 - 2*x^6 + x^8 + 2*x^9 - 2*x^10 + ...
G.f. = q + q^4 - 2*q^7 + 2*q^13 - 3*q^16 - 2*q^19 + q^25 + 2*q^28 - ...

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. A033687, A121361, A122861, A123884, A129451, A131961, A131962, A131963, A131964.

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

Expansion of q^(-1/3) * eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.
a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = A122861(n). a(4*n + 2) = -2 * A121361(n). a(4*n + 3) = 0.
a(8*n) = A131961(n). a(8*n + 1) = A097195(n). a(8*n + 2) = -2 * A131962(n). a(8*n + 4) = 2 * A131963(n). a(8*n + 6) = -2 * A131964(n).
a(16*n + 1) = A123884(n). a(16*n + 5) = -3 * A033687(n). a(16*n + 9) = 2 * A121361(n). a(16*n + 13) = 0.

A246752 Expansion of phi(-x) * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 02 2014

sign

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^4 + 3*x^5 - 2*x^6 + x^8 - 2*x^9 - 2*x^10 + ...
G.f. = q - q^4 - 2*q^7 + 2*q^13 + 3*q^16 - 2*q^19 + q^25 - 2*q^28 + ...

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. A002448, A089801, A121361, A122861, A123884, A129451, A246650, A246838.

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

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

Expansion of phi(-x) * f(x^1, x^5) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q^4) * eta(q^6)) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246838.
a(n) = (-1)^n * A246650(n).
Convolution of A002448 and A089801.
a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = - A122861(n). a(4*n + 2) = - 2 * A121361(n). a(4*n + 3) = 0.

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

A229143 Expansion of (b(q^3) - b(q)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 23 2013

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Rogers and Zudilin (2011) page 6: "This identity can be verified by eliminating b(q) with b(q^{1/3}) - b(q) = 3c(q^3) - c(q)."
The zeros of the g.f. A(q) where q = exp(2 Pi i t) are of the form t = (m/2 + sqrt(-3)/18) / n where m is an odd integer and n is in A004611. For example, (1/2 + sqrt(-3)/18) / 1, (1/2 + sqrt(-3)/18) / 7, (5/2 + sqrt(-3)/18) / 13.

			G.f. = q - 3*q^3 + q^4 + 2*q^7 - 3*q^12 + 2*q^13 + q^16 + 2*q^19 - 6*q^21 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
M. Rogers and W. Zudilin, On the Mahler measure of 1 + X + 1/X + Y + 1/Y, arXiv:1102.1153 [math.NT], 2011.

Cf. A004611, A033687, A097195, A121361, A123884, A259024.

A := Basis( ModularForms( Gamma1(27), 1), 85); A[2] - 3*A[4] + A[5] + 2*A[8] - 3*A[13] + 2*A[14] + A[15]; /* Michael Somos, Jun 16 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3]^4 - QPochhammer[ q^9] QPochhammer[ q]^3) / (3 QPochhammer[ q^3] QPochhammer[ q^9]), {q, 0, n}]; (* Michael Somos, Jun 16 2015 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9]^4 - 3 q^2 QPochhammer[ q^3] QPochhammer[ q^27]^3) / (QPochhammer[ q^3] QPochhammer[ q^9]), {q, 0, n}]; (* Michael Somos, Jun 16 2015 *)
f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1 + (-1)^e) / 2]; f[3, 1] = -3; f[3, e_] := 0; a[n_] := Times @@ f @@@ FactorInteger[n]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, Sep 04 2023 *)

{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==3, -3 * (e==1), p%3==1, e+1, !(e%2))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^9 + A)^4 - 3 * x^2 * eta(x^3 + A) * eta(x^27 + A)^3) / (eta(x^3 + A) * eta(x^9 + A)), n))};

Expansion of c(q^3) / 3 - c(q^9) in powers of q where c() is a cubic AGM theta function.
Expansion of (a(q) - 4*a(q^3) + 3*a(q^9)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of (eta(q^3)^4 - eta(q)^3 * eta(q^9)) / (3 * eta(q^3) * eta(q^9)) in powers of q.
a(n) is multiplicative with a(3) = -3, a(3^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1 + (-1)^e) / 2 if p == 2 (mod 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(3*n + 2) = a(4*n + 2) = a(9*n) = a(9*n + 6) = 0. a(3*n + 1) = A033687(n). a(9*n + 3) = -3 * A033687(n).
From Michael Somos, Jun 16 2015: (Start)
a(4*n) = a(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(n) = Sum_{d|n} A259024(n/d) * [ 0, 1, 0, -2, 0, 1][mod(d, 6) + 1]. (End)

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

A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 6, 4, 0, 0, 6, 8, 6, 4, 0, 0, 0, 8, 12, 0, 0, 0, 6, 8, 0, 8, 0, 0, 6, 4, 12, 4, 0, 0, 0, 8, 6, 0, 0, 0, 0, 8, 12, 8, 0, 0, 12, 8, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 12, 8, 0, 0, 0, 8, 12, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 8, 12, 4, 0, 0, 12, 8, 0, 4, 0, 0

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 + 4*x + 6*x^2 + 4*x^3 + 6*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + 8*x^13 + ...

Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.

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

A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];

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

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

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

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

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

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*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^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].
Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 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 A253625.
a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) 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} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).
G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.
a(n) = (-1)^n * A244339(n). a(2*n) = A004016(n). a(2*n + 1) = 4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = 4 * A097195(n). a(6*n + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.
a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023

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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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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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A246650 Expansion of phi(x) * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A246752 Expansion of phi(-x) * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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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.

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A229143 Expansion of (b(q^3) - b(q)) / 3 in powers of q where b() is a cubic AGM theta function.

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