A316662 Expansion of f(x, x^2) * psi(x^3)^3 in powers of x where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

1, 1, 1, 3, 3, 4, 3, 4, 6, 4, 7, 7, 7, 9, 10, 7, 7, 9, 12, 12, 9, 16, 13, 15, 13, 12, 13, 16, 19, 13, 18, 16, 19, 21, 19, 20, 21, 21, 24, 19, 22, 24, 22, 27, 25, 21, 21, 27, 30, 32, 27, 28, 30, 37, 25, 31, 34, 28, 36, 28, 31, 28, 31, 48, 36, 36, 38, 31, 42, 37

Offset

0,4

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-7/6) * eta(q^2) * eta(q^6)^5 / (eta(q) * eta(q^3)) in powers of q.

Expansion of b(x^2) * c(x^2)^2 / sqrt(b(x) * c(x)/3) in powers of x where b(), c() are cubic AGM theta functions.

Expansion of psi(x^3) * f(-x^6)^3 / chi(-x) = f(-x^6)^4 / (chi(-x) * chi(-x^3)) = f(-x^6)^5 /(f(-x^3) * chi(-x)) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 - x^(6*k))^4.

Example

G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + ...
G.f. = q^7 + q^13 + q^19 + 3*q^25 + 3*q^31 + 4*q^37 + 3*q^43 + 4*q^49 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)), n))};
(Magma) Basis( ModularForms( Gamma0(36), 2), 427) [8];

Crossrefs

Sequence in context: A175928 A161357 A138372 * A123708 A350501 A102302

Adjacent sequences: A316659 A316660 A316661 * A316663 A316664 A316665

Keyword

nonn

Author

Michael Somos, Jul 09 2018

Status

approved