A123630 Expansion of q * (chi(-q^3) * chi(-q^5)) / (chi(-q) * chi(-q^15))^2 in powers of q where chi() is a Ramanujan theta function.

1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 106, 137, 175, 222, 280, 352, 439, 546, 676, 834, 1024, 1253, 1528, 1857, 2250, 2718, 3276, 3936, 4718, 5640, 6728, 8006, 9507, 11266, 13324, 15726, 18526, 21786, 25574, 29970, 35064, 40961, 47774, 55638

Offset: 1

Author

Michael Somos, Oct 03 2006

Keywords

• nonn

Comments

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

Examples

			G.f. = q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 10*q^6 + 14*q^7 + 20*q^8 + 27*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A094023, A133098.

Programs

Mathematica

f[q_] := QPochhammer[-q, q^2]; A123630[n_] := SeriesCoefficient[q*(f[-q^3]*f[-q^5])/(f[-q]*f[-q^15])^2, {q, 0, n}]; Table[A123630[n], {n, 0, 50}] (* G. C. Greubel, Oct 17 2017 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x*O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A)^2 / (eta(x + A)^2 * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A)^2), n))};

Formula

Euler transform of period 30 sequence [ 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 -v - u*v * (4 + 2*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133098.

G.f.: x * Product_{k>0} (1 + x^k) * (1 + x^(15*k)) * P(30, x^k) where P(n, x) is n th cyclotomic polynomial.

a(n) = A094023(n) unless n=0. Convolution inverse is A133098.

a(n) ~ exp(2*Pi*sqrt(2*n/15)) / (2^(7/4) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017