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

1,3

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 = 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

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A123629.

Sequence in context: A062464 A053270 A261154 * A003412 A038348 A239467

Adjacent sequences: A233690 A233691 A233692 * A233694 A233695 A233696

Keyword

nonn

Author

Michael Somos, Dec 14 2013

Status

approved