A233037 Expansion of 3 * q^(1/3) * phi(q) * psi(q^6) / c(q) in powers of x where phi(), psi() are Ramanujan theta functions and c() is a cubic AGM theta function.

1, 1, -3, 1, 5, -10, 4, 17, -31, 9, 46, -79, 21, 112, -183, 46, 249, -396, 98, 521, -815, 193, 1041, -1599, 373, 1998, -3031, 696, 3708, -5567, 1262, 6694, -9955, 2233, 11788, -17393, 3872, 20313, -29771, 6572, 34342, -50016, 10973, 57065, -82654, 18030

Offset

0,3

Comments

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).

Links

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

Formula

Expansion of q^(-5/12) * eta(q^2)^5 * eta(q^12)^2 / (eta(q) * eta(q^4)^2 * eta(q^3)^3 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 1, -4, 4, -2, 1, 0, 1, -2, 4, -4, 1, 0, ...].

a(n) = A182032(12*n + 5).

Example

G.f. = 1 + x - 3*x^2 + x^3 + 5*x^4 - 10*x^5 + 4*x^6 + 17*x^7 - 31*x^8 + ...
G.f. = q^5 + q^17 - 3*q^29 + q^41 + 5*q^53 - 10*q^65 + 4*q^77 + 17*q^89 + ...

Mathematica

eta[x_] := x^(1/24)*QPochhammer[x]; A233037[n_] := SeriesCoefficient[q^(-5/12)*eta[q^2]^5* eta[q^12]^2/(eta[q]*eta[q^4]^2*eta[q^3]^3*eta[q^6]), {q, 0, n}];
Table[A233037[n], {n, 0, 50}] (* G. C. Greubel, Aug 10 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^3 + A)^3 * eta(x^6 + A)), n))}

Crossrefs

Cf. A182032.

Sequence in context: A103327 A177463 A065229 * A275999 A286910 A093905

Adjacent sequences: A233034 A233035 A233036 * A233038 A233039 A233040

Keyword

sign

Author

Michael Somos, Dec 03 2013

Status

approved