A132091 Expansion of psi(x^3) * chi(-x^9) / f(-x^2) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

1, 0, 1, 1, 2, 1, 3, 2, 5, 3, 7, 5, 10, 7, 14, 11, 20, 15, 27, 22, 37, 30, 49, 42, 66, 56, 86, 75, 113, 99, 146, 131, 189, 170, 241, 221, 308, 283, 389, 363, 492, 460, 616, 583, 771, 732, 958, 918, 1189, 1143, 1467, 1421, 1807, 1756, 2215, 2166, 2711, 2658, 3303, 3256

Offset

0,5

Comments

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

Also number of partitions of n into parts not divisible by 3 with every part repeated at least twice. Conjectured by R. H. Hardin, Jun 06 2009, proved by Max Alekseyev, Jun 06 2009.

The number of partitions of n into parts not divisible by 3 with every part repeated at least twice has g.f. f(x) = Product_{k>=1} (1 + x^(2k) + x^(3*k) + ...) = Product_{k>=1} (1/(1-x^k) - x^k) = Product_{k>=1} (1 - x^k + x^(2*k)) / (1 - x^k). Excluding parts divisible by 3, we have: f(x) / f(x^3) = Product_{k>=1} (1 - x^k + x^(2*k)) * (1 - x^(3*k)) / (1 - x^k) / (1 - x^(3*k) + x^(6*k)) = Product_{k>=1} (1 - x^k + x^(2*k)) * (1 + x^k + x^(2*k)) / (1 - x^(3*k) + x^(6*k)) = Product_{k>=1} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k)), which matches the definition of this sequence. - Max Alekseyev, Jun 06 2009

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 903 terms from R. H. Hardin)

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 q^(1/12) * eta(q^6)^2 * eta(q^9) / ( eta(q^2) * eta(q^3) * eta(q^18)) in powers of q.

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

G.f.: Product_{k>0} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k)).

G.f.: Sum_{k>=0} Product_{0<i<=k} x^(4*i - 2) * (1 - x^(6*i - 3)) / ( (1 - x^(2*i - 1)) * (1 - x^(4*i - 2)) * (1 - x^(4*i))).

a(2*n - 1) = A000701(n). a(2*n) = A027340(n) = - Michael Somos, Aug 25 2015

a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (2^(3/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 3*x^9 + ...
G.f. = 1/q + q^23 + q^35 + 2*q^47 + q^59 + 3*q^71 + 2*q^83 + 5*q^95 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8) QPochhammer[ -x^9, x^9] QPochhammer[ x^2]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^2 * (1-x^(9*k)) / ( (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(18*k))) , {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

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

Crossrefs

Cf. A000701, A027340.

Sequence in context: A161255 A008731 A114209 * A262090 A239881 A051792

Adjacent sequences: A132088 A132089 A132090 * A132092 A132093 A132094

Keyword

nonn

Author

Michael Somos, Aug 09 2007

Extensions

Edited by N. J. A. Sloane, Jun 07 2009

Status

approved