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A123655 Expansion of q * psi(q^8) / phi(-q) in powers of q where psi(), phi() are Ramanujan theta functions. 5
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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

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Last modified September 15 22:10 EDT 2019. Contains 327088 sequences. (Running on oeis4.)