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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
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 (list; graph; refs; listen; history; text; internal format)
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

R. H. Hardin and 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.

M. 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

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Last modified November 22 08:57 EST 2017. Contains 295076 sequences.