A033687 -id:A033687 - cp's OEIS frontend

A193426 Expansion of (a(q^2) + a(q^3) - 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

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

Offset: 1

Michael Somos, Jul 27 2011

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. = q^2 + q^3 - q^6 + q^8 + q^9 + q^12 + 2*q^14 - q^18 + 2*q^21 - q^24 + ...

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. A033687, A033762, A093829.

a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^9]^2 / EllipticTheta[ 2, 0, q^3], {q, 0, 2 n}];

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

{a(n) = my(A); if( n<2, 0, n = n-2; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^4 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2), n))};

Expansion of (b(q^6)^2 / b(q^3) - b(q^2)) / 3 = (c(q^6) / c(q^3)) * (c(q^3) + c(q^6)) / 3 = q^2 * psi(q) * psi(q^9)^2 / psi(q^3) in powers of q where b(), c() are cubic AGM functions and psi() is a Ramanujan theta function.
Expansion of eta(q^2)^2 * eta(q^3) * eta(q^18)^4 / (eta(q) * eta(q^6)^2 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 1, -1, 0, -1, 1, 0, 1, -1, 2, -1, 1, 0, 1, -1, 0, -1, 1, -2, ...].
Moebius transform is period 18 sequence [ 0, 1, 1, -1, 0, -3, 0, 1, 0, -1, 0, 3, 0, 1, -1, -1, 0, 0, ...].
a(3*n) = A093829(n). a(6*n) = -A093829(n). a(6*n + 2) = A033687(n). A(6*n + 3) = A033762(n). a(3*n + 1) = a(6*n + 5) = 0. a(4*n) = a(n).

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

A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, May 06 2015

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. = q + q^2 + 2*q^3 + q^4 + 2*q^6 + 2*q^7 + q^8 + 2*q^9 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A035178, A248897.

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

a[ n_] := If[ n < 1, 0, Sum[ { 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0}[[Mod[ d, 18, 1]]], { d, Divisors[ n]}]];

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1][d%18 + 1]))};

{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, 1, p==3, 2, p%6==1, e+1, 1-e%2)))};

Expansion of (b(q^2)^2 / b(q) + b(q^6)^2 / b(q^3) - 2) / 3 in powers of q where b() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3) + psi(q^3)^3 / psi(q^9) - 2) / 3 in powers of q where psi() is a Ramanujan theta function.
Moebius transform is period 18 sequence [ 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2 if 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} (x^k + x^(3*k)) / (1 + x^(2*k))^2 + (x^(3*k) + x^(9*k)) / (1 + x^(6*k))^2.
a(2*n) = a(n). a(3*n) = 2 * A035178(n). a(3*n + 1) = A033687(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 22 2023

A286952 Expansion of Product_{j>=1} (1 - x^j)/(1 - x^(3*j))^3.

Original entry on oeis.org

1, -1, -1, 3, -3, -2, 9, -8, -6, 22, -19, -13, 50, -42, -29, 104, -86, -57, 209, -170, -113, 398, -320, -208, 737, -586, -380, 1320, -1041, -666, 2311, -1808, -1152, 3949, -3069, -1938, 6629, -5120, -3221, 10920, -8390, -5242, 17724, -13549, -8435, 28342

Offset: 0

Seiichi Manyama, May 17 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

3rd column of A286950.

Cf. A033687.

A115235 Expansion of eta(q)^2eta(q^9)eta(q^18)/(eta(q^2)*eta(q^3)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 17 2006

			q - 2*q^2 + q^4 + 2*q^7 - 2*q^8 + 2*q^13 - 4*q^14 + q^16 + 2*q^19 + ...

Cf. A033687.

f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1+3*(-1)^e)/2; f[3, e_] := 0; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 04 2023 *)

{a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^9+A)*eta(x^18+A)/eta(x^2+A)/eta(x^3+A), n))}

Euler transform of period 18 sequence [ -2, -1, -1, -1, -2, 0, -2, -1, -2, -1, -2, 0, -2, -1, -1, -1, -2, -2, ...].
Moebius transform is period 18 sequence [1, -3, -1, 3, -1, 3, 1, -3, 0, 3, -1, -3, 1, -3, 1, 3, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1+3*(-1)^e)/2, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3n+1) = A033687(n), a(6n+2) = -2*A033687(n), a(3n) = a(6n+5) = 0. a(4n) = a(n).
G.f.: Sum_{k} x^(3k+1)*(1-2*x^(3k+1))/(1-x^(18k+6)).

A131140 Counts 3-wild partitions. In general p-wild partitions of n are defined so that they are in bijection with geometric equivalence classes of degree n algebra extensions of the p-adic field Q_p. Here two algebra extensions are equivalent if they become isomorphic after tensoring with the maximal unramified extension of Q_p.

Original entry on oeis.org

1, 1, 2, 9, 11, 19, 83, 99, 172, 1100, 1244, 2250, 8687, 10683, 18173, 67950, 82785, 140825, 665955, 780030, 1367543, 4867750, 6027860, 10149291, 35453711, 43581422

Offset: 0

David P. Roberts (roberts(AT)morris.umn.edu), Jun 19 2007

In general, the number of p-wild partitions of n is equal to the number of partitions of n if and only if n

			a(3) = 9, since there are four quadratic algebras over Q_3 up to geometric equivalence, namely the unramified algebra Q_3 times Q_3 times Q_3, the tamely ramified algebras Q_3 times Q_3[x]/(x^2-3) and two, two and three wildly ramified algebras with discriminants 3^3, 3^4 and 3^5 respectively.

David P. Roberts, Wild Partitions and Number Theory. Journal of Integer Sequences, Volume 10, Issue 6, Article 07.6.6, (2007).

Cf. A000041, A033687, A131139.

The generating function is Product_{j>=0} theta_3(2^((3^j-1)/2)*x)^(3^j) where theta_3(y) is the generating function for 3-cores A033687. [This appears to be incorrect - Joerg Arndt, Apr 06 2013]

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 4, 2, -4, 0, 4, -8, 4, 2, 0, 0, 2, -8, 8, 0, -4, 0, 4, -8, 0, 4, 0, 0, 4, -4, 8, 2, -8, 0, 0, -8, 4, 0, 0, 0, 2, -8, 8, 4, 0, 0, 8, -8, 0, 0, 0, 0, 2, -12, 4, 0, -8, 0, 4, 0, 8, 4, 0, 0, 0, -8, 8, 4, -4, 0, 0, -8, 0, 0, 0, 0, 4, -8, 8, 2, -8, 0, 8, -8, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -16, 0, 4, 0, 0, 4, -8

Offset: 0

Michael Somos, Jul 29 2011

sign

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

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

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, A033762, A097195, A112604, A112605.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

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

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)), n))};

Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).

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

A258724 Expansion of f(-x)^11 / f(-x^3) + 27 * x * f(-x^3)^11 / f(-x) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 16, 71, 0, -337, 256, -601, 0, 625, 1136, 194, 0, -529, 0, -3214, 0, 2640, -5392, 0, 0, 7199, 4096, 2903, 0, -1249, -9616, 4679, 0, 0, 0, -23927, 0, 9071, 10000, -19849, 0, 22034, 18176, 0, 0, 14641, 3104, -10942, 0, -42671, 0, 24359, 0, 0, -8464, -42121

Offset: 0

Michael Somos, Jun 08 2015

sign

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

This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731.

			G.f. = 1 + 16*x + 71*x^2 - 337*x^4 + 256*x^5 - 601*x^6 + 625*x^8 + ...
G.f. = q + 16*q^4 + 71*q^7 - 337*q^13 + 256*q^16 - 601*q^19 + ...

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. A000731, A030206, A033687, A130539.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^11 / QPochhammer[ x^3] + 27 x QPochhammer[ x^3]^11 / QPochhammer[ x], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^11 / eta(x^3 + A) + 27 * x * eta(x^3 + A)^11 / eta(x + A), n))};

{a(n) = my(A, p, e, x, y, a0, a1); n = 3*n + 1; if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3, 0, p%3==2, if( e%2, 0, p^(2*e)), for( x=1, sqrtint(4*p\27), if( issquare(4*p - 27*x^2, &y), break)); y = (y^2 - 2*p)^2 - 2*p^2; a0=1; a1=y; for( i=2, e, x = y*a1 - p^4*a0; a0=a1; a1=x); a1)))};

Expansion of q^(-1/3) * (eta(q)^11 / eta(q^3) + 27 * eta(q^3)^11 / eta(q)) in powers of q.
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(2*e) if p == 2 (mod 3), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 3) where b(p) = (y^2 - 2*p)^2 - 2*p^2, 4*p = y^2 + 27*x^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(5/2) (t/i)^5 f(t) where q = exp(2 Pi i t).
a(4*n + 3) = 0.

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A193426 Expansion of (a(q^2) + a(q^3) - 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

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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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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.

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A255648 Expansion of (a(q) + a(q^2) + a(q^3) + a(q^6) - 4) / 6 in powers of q where a() is a cubic AGM theta function.

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A286952 Expansion of Product_{j>=1} (1 - x^j)/(1 - x^(3*j))^3.

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A115235 Expansion of eta(q)^2*eta(q^9)*eta(q^18)/(eta(q^2)*eta(q^3)) in powers of q.

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A115235 Expansion of eta(q)^2eta(q^9)eta(q^18)/(eta(q^2)*eta(q^3)) in powers of q.