A262175 Expansion of chi(x) * psi(x^6) * phi(-x^30) / (f(-x^4) * psi(x^5)) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

1, 1, 0, 1, 2, 1, 1, 3, 4, 4, 4, 6, 8, 8, 8, 11, 16, 17, 17, 23, 31, 32, 32, 42, 54, 56, 59, 77, 94, 99, 106, 129, 156, 167, 178, 214, 257, 276, 295, 350, 416, 445, 474, 559, 652, 698, 752, 877, 1012, 1089, 1174, 1349, 1542, 1662, 1792, 2042, 2327, 2512, 2706

Offset

0,5

Comments

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

Links

Vaclav Kotesovec, 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^(1/12) * eta(q^2)^2 * eta(q^5) * eta(q^12)^2 * eta(q^30)^2 / (eta(q) * eta(q^4)^2 * eta(q^6) * eta(q^10)^2 * eta(q^60)) in powers of q.

Euler transform of a period 60 sequence.

a(n) = A139632(3*n).

a(n) ~ exp(Pi*sqrt(3*n/10)) / (2^(5/4) * 3^(3/4) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 + x + x^3 + 2*x^4 + x^5 + x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^-1 + q^11 + q^35 + 2*q^47 + q^59 + q^71 + 3*q^83 + 4*q^95 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-1/8) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^3] EllipticTheta[ 4, 0, x^30] / (QPochhammer[ x^4] EllipticTheta[ 2, 0, x^(5/2)]), {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^5 + A) * eta(x^12 + A)^2 * eta(x^30 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^10 + A)^2 * eta(x^60 + A)), n))};

Crossrefs

Cf. A139632.

Sequence in context: A302097 A307277 A210691 * A278028 A124424 A057044

Adjacent sequences: A262172 A262173 A262174 * A262176 A262177 A262178

Keyword

nonn

Author

Michael Somos, Sep 13 2015

Status

approved