A228072 Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 8, -10, 16, 37, -40, -50, -80, -30, 40, 128, 48, -25, 80, -34, 320, -320, -160, 310, -400, 410, 152, -370, -416, -87, -240, -410, 400, 320, -200, 30, 592, 500, 776, 384, 400, -630, -200, -640, -1120, -359, 552, 300, -272, -326, -800, 2560, -400, -110

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

Hossein Movasati, Younes Nikdelan, Product formulas for weight two newforms, arXiv:1803.01414 [math.NT], 2018.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * ((eta(q^2)^5 / eta(q^4))^2 + 8 * (eta(q^4)^5 / eta(q^2))^2) in powers of q.

Expansion of q^(-1/2) * (eta(q^2)^12 + 8 * eta(q^4)^12) / ( eta(q^2) * eta(q^4) )^2 in powers of q.

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^2 (t / i)^4 f(t) where q = exp(2 Pi i t).

a(2*n) = A227695(n). a(2*n + 1) = 8 * A227317(n).

If F(x) is the g.f. for A002171, then A(x) * F(x^2) = B(x) the g.f. for A227239. - Michael Somos, Jan 08 2015

Example

G.f. = 1 + 8*x - 10*x^2 + 16*x^3 + 37*x^4 - 40*x^5 - 50*x^6 - 80*x^7 - 30*x^8 + ...
G.f. = q + 8*q^3 - 10*q^5 + 16*q^7 + 37*q^9 - 40*q^11 - 50*q^13 - 80*q^15 - 30*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^12 + 8 x QPochhammer[ x^4]^12) / (QPochhammer[ x^2] QPochhammer[ x^4])^2, {x, 0, n}];

Prog

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

Crossrefs

Cf. A002171, A227239, A227317, A227695.

Sequence in context: A355568 A289687 A155966 * A038209 A319293 A061908

Adjacent sequences: A228069 A228070 A228071 * A228073 A228074 A228075

Keyword

sign

Author

Michael Somos, Sep 02 2013

Status

approved