A128129 Expansion of (chi(-q^3)/ chi^3(-q) -1)/3 in powers of q where chi() is a Ramanujan theta function.

1, 2, 4, 7, 12, 20, 32, 50, 76, 114, 168, 244, 350, 496, 696, 967, 1332, 1820, 2468, 3324, 4448, 5916, 7824, 10292, 13471, 17548, 22756, 29384, 37788, 48408, 61784, 78578, 99600, 125838, 158496, 199036, 249230, 311224, 387608, 481506, 596676

Offset: 1

Author

Michael Somos, Feb 15 2007

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10001
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S76.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

A128128(n)=3*a(n) if n>0.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(18*k))*(1 - x^(18*k - 3))*(1 - x^(18*k - 15))/((1 - x^(2*k - 1))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 11 2017 *)

PARI

{a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^2+A)* eta(x^3+A)* eta(x^18+A)^2/ (eta(x^6+A)* eta(x^9+A)* eta(x+A)^2), n))}

Formula

Expansion of (eta(q^2)^3* eta(q^3)/ (eta(q)^3* eta(q^6)) -1)/3 in powers of q.

Euler transform of period 18 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v -2*v^2 -4*u*v -6*u*v^2.

G.f. A(x) satisfies 0=f(A(x), A(x^3)) where f(u, v)= u^3 -v* (1+3*v+3*v^2)* (1+6*u+12*u^2).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(7/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Jan 12 2017