A233693 Expansion of q * psi(-q) * chi(-q^6) * psi(-q^9) / (phi(-q) * phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 48, 60, 75, 92, 114, 140, 170, 208, 252, 304, 366, 439, 526, 626, 744, 884, 1044, 1232, 1451, 1704, 1998, 2336, 2730, 3182, 3700, 4300, 4986, 5772, 6672, 7700, 8876, 10212, 11736, 13472, 15438, 17673, 20207, 23076

Offset: 1

Author

Michael Somos, Dec 14 2013

Keywords

• nonn

Comments

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

Examples

			G.f. = q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 8*q^7 + 11*q^8 + 14*q^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A123629.

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(6*k)) * (1-x^(9*k)) * (1+x^(18*k))^2 / ((1-x^k) * (1-x^(12*k)) * (1-x^(18*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
QP := QPochhammer; A233693[n_]:= SeriesCoefficient[QP[q^4]*QP[q^6] *QP[q^9]*QP[q^36]^2/(QP[q]* QP[q^12]*QP[q^18]^3), {q, 0, n}]; Table[A233693[n], {n, 0, 50}] (* G. C. Greubel, Dec 25 2017 *)

PARI

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)^2 / (eta(x + A) * eta(x^12 + A) * eta(x^18 + A)^3), n))}

Formula

Expansion of eta(q^4) * eta(q^6) * eta(q^9) * eta(q^36)^2 / (eta(q) * eta(q^12) * eta(q^18)^3) in powers of q.

Euler transform of period 36 sequence [ 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, ...].

a(2*n) = A123629(n).

a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015