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

1,2

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

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

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

Example

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 + ...

Mathematica

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

Prog

(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))};

Crossrefs

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

Sequence in context: A004402 A015128 A208605 * A084683 A271493 A118544

Adjacent sequences: A123652 A123653 A123654 * A123656 A123657 A123658

Keyword

nonn

Author

Michael Somos, Oct 04 2006

Status

approved