A123655 Expansion of q * psi(q^8) / phi(-q) in powers of q where psi(), phi() are Ramanujan theta functions.

1, 2, 4, 8, 14, 24, 40, 64, 101, 156, 236, 352, 518, 752, 1080, 1536, 2162, 3018, 4180, 5744, 7840, 10632, 14328, 19200, 25591, 33932, 44776, 58816, 76918, 100176, 129952, 167936, 216240, 277476, 354864, 452392, 574958, 728568, 920600, 1160064

Offset: 1

Author

Michael Somos, Oct 04 2006

Keywords

• nonn

Comments

Ramanujan theta functions: phi(q) (A000122), psi(q) (A010054).

Number 12 of the 14 eta-quotients listed in Table 2 of Moy 2013. - Michael Somos, Sep 19 2013

Examples

			G.f. = q + 2*q^2 + 4*q^3 + 8*q^4 + 14*q^5 + 24*q^6 + 40*q^7 + 64*q^8 + 101*q^9 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Richard Moy, Congruences among power series coefficients of modular forms, arXiv:1309.4320
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A007096, A185338, A093160, A107035, A185338.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^4] / EllipticTheta[ 4, 0, q] / 2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)), n))};

Formula

Expansion of eta(q^2) * eta(q^16)^2 / (eta(q)^2 * eta(q^8)) in powers of q.

Euler transform of period 16 sequence [ 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 2*v).

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/8 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A185338.

a(n) is odd iff n is an odd square. If n>2 is a power of 2 then the highest power of 2 dividing a(n) is (n/2)^3. - Michael Somos, Feb 18 2007

4 * a(n) = A007096(n) unless n=0. -(-1)^n * a(n) = A208605(n). Convolution inverse of A185338.

G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k)) * (1 + x^(4*k)) * (1 + x^(8*k))^2. Michael Somos, Sep 19 2013

a(2*n) = 2 * A107035(n). a(2*n + 1) = A093160(n). - Michael Somos, Sep 19 2013

a(n) ~ exp(sqrt(n)*Pi) / (2^(9/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017