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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.

%I A132091 #28 Feb 16 2025 08:33:06
%S A132091 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,
%T A132091 75,113,99,146,131,189,170,241,221,308,283,389,363,492,460,616,583,
%U A132091 771,732,958,918,1189,1143,1467,1421,1807,1756,2215,2166,2711,2658,3303,3256
%N A132091 Expansion of psi(x^3) * chi(-x^9) / f(-x^2) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%C A132091 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A132091 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.
%C A132091 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
%H A132091 Alois P. Heinz, <a href="/A132091/b132091.txt">Table of n, a(n) for n = 0..10000</a> (first 903 terms from R. H. Hardin)
%H A132091 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.
%H A132091 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132091 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132091 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.
%F A132091 Euler transform of period 18 sequence [ 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, ...].
%F A132091 G.f.: Product_{k>0} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k)).
%F A132091 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))).
%F A132091 a(2*n - 1) = A000701(n). a(2*n) = A027340(n) = - _Michael Somos_, Aug 25 2015
%F A132091 a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (2^(3/4) * 3^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A132091 G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 3*x^9 + ...
%e A132091 G.f. = 1/q + q^23 + q^35 + 2*q^47 + q^59 + 3*q^71 + 2*q^83 + 5*q^95 + ...
%t A132091 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 *)
%t A132091 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 *)
%o A132091 (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))};
%Y A132091 Cf. A000701, A027340.
%K A132091 nonn
%O A132091 0,5
%A A132091 _Michael Somos_, Aug 09 2007
%E A132091 Edited by _N. J. A. Sloane_, Jun 07 2009