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

1,2

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

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

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 *)

Prog

(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))}

Crossrefs

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

Sequence in context: A036372 A132218 A101230 * A014968 A289115 A342528

Adjacent sequences: A128126 A128127 A128128 * A128130 A128131 A128132

Keyword

nonn

Author

Michael Somos, Feb 15 2007

Status

approved