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

Author

Michael Somos, Sep 13 2015

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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

Crossrefs

Cf. A139632.

Programs

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}];

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))};

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