A260546 Expansion of phi(-x^3) * psi(-x^3) / phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions

1, 4, 12, 29, 64, 132, 258, 484, 876, 1539, 2636, 4416, 7256, 11716, 18624, 29190, 45164, 69060, 104457, 156416, 232044, 341256, 497804, 720648, 1035792, 1478720, 2097612, 2957590, 4146284, 5781120, 8018821, 11067748, 15203904, 20791590, 28310028, 38387556

Offset

0,2

Comments

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

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^(-3/8) * eta(q^2)^2 * eta(q^3)^3 * eta(q^12) / (eta(q)^4 * eta(q^6)^2) in powers of q.

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

a(n) = A001935(3*n + 1).

a(n) ~ exp(Pi*sqrt(3*n/2)) / (2^(11/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 + 4*x + 12*x^2 + 29*x^3 + 64*x^4 + 132*x^5 + 258*x^6 + 484*x^7 + ...
G.f. = q^3 + 4*q^11 + 12*q^19 + 29*q^27 + 64*q^35 + 132*q^43 + 258*q^51 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A001935.

Sequence in context: A128563 A227085 A192978 * A062421 A036889 A036895

Adjacent sequences: A260543 A260544 A260545 * A260547 A260548 A260549

Keyword

nonn

Author

Michael Somos, Jul 28 2015

Status

approved