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A131126 Expansion of (phi(q^2) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function. 8
1, 4, 16, 48, 128, 312, 704, 1504, 3072, 6036, 11488, 21264, 38400, 67864, 117632, 200352, 335872, 554952, 904784, 1457136, 2320128, 3655296, 5702208, 8813472, 13504512, 20523996, 30952544, 46340832, 68901888, 101777112, 149403264, 218016640, 316342272 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, 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

FORMULA

Expansion of ((phi(q) / phi(-q))^2 + 1) / 2 in powers of q where phi() is a Ramanujan theta function.

Expansion of (eta(q^4)^5 / (eta(q)^2 * eta(q^2) * eta(q^8)^2))^2 in powers of q.

Euler transform of period 8 sequence [ 4, 6, 4, -4, 4, 6, 4, 0, ...].

a(n) = 4 * A107035(n) unless n=0. 2 * a(n) = A014969(n) unless n=0.

a(n) ~ exp(sqrt(2*n)*Pi) / (16 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + (1/8)*sqrt(8 + 6*sqrt(2)). - Simon Plouffe, Mar 04 2021

EXAMPLE

G.f. = 1 + 4*q + 16*q^2 + 48*q^3 + 128*q^4 + 312*q^5 + 704*q^6 + 1504*q^7 + ...

MATHEMATICA

nmax = 50; CoefficientList[Series[Product[((1 - x^(4*k))^5 / ((1 - x^k)^2 * (1 - x^(2*k)) * (1 - x^(8*k))^2))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^5)^-2, n))};

CROSSREFS

Cf. A014969, A107035.

Sequence in context: A223839 A071009 A210066 * A159964 A058922 A215723

Adjacent sequences:  A131123 A131124 A131125 * A131127 A131128 A131129

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 15 2007

STATUS

approved

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Last modified July 23 22:20 EDT 2021. Contains 346265 sequences. (Running on oeis4.)