%I #29 Jun 06 2018 18:31:25

%S 1,2,3,4,5,8,11,16,21,26,34,44,58,74,93,116,143,178,221,272,332,402,

%T 487,588,710,854,1021,1216,1444,1714,2031,2400,2826,3318,3888,4552,

%U 5322,6208,7224,8388,9726,11264,13028,15044,17339,19952,22930,26324,30186

%N McKay-Thompson series of class 36g for the Monster group.

%C Number of partitions of n into distinct parts prime to 3, with 2 types of each part.

%C This is also the number of partitions of n into parts with 2 types congruent to 1 or 5 mod(6).

%H G. C. Greubel, <a href="/A103262/b103262.txt">Table of n, a(n) for n = 0..1000</a>

%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F G.f.: product_{k>0}((1+x^k)/(1+x^(3k)))^2= 1/product_{k>0}((1-x^(6k-1))*(1-x^(6k-5)))^2.

%F Expansion of q^(1/6)(eta(q^2)eta(q^3)/(eta(q)eta(q^6)))^2 in powers of q.

%F Euler transform of period 6 sequence [2, 0, 0, 0, 2, 0, ...]. - _Michael Somos_, Sep 10 2005

%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Sep 01 2015

%e E.g., a(5)=8 because we have 5,5*,41,41*,4*1,4*1*,22*1,22*1* with all parts prime to 3. The parts congruent to 1,5 mod(6) are 5, 5*, 11111, 11111*, 1111*1*, 111*1*1*, 11*1*1*1*, 1*1*1*1*1*.

%e T36g = 1/q + 2*q^5 + 3*q^11 + 4*q^17 + 5*q^23 + 8*q^29 + 11*q^35 + ...

%p series(product((1+x^k)^2/(1+x^(3*k))^2,k=1..100),x=0,100);

%t CoefficientList[ Series[ Product[(1 + x^k)^2/(1 + x^(3k))^2, {k, 60}], {x, 0, 50}], x] (* _Robert G. Wilson v_, Feb 22 2005 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/6)(eta[q^2]eta[q^3]/(eta[q]eta[q^6]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 06 2018 *)

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)/eta(x+A)/eta(x^6+A))^2, n))} /* _Michael Somos_, Sep 10 2005 */

%Y Cf. A003105.

%K nonn

%O 0,2

%A _Noureddine Chair_, Feb 21 2005

%E More terms from _Robert G. Wilson v_, Feb 22 2005