%I #30 Jan 06 2020 13:32:32
%S 1,1,2,1,3,3,4,4,6,8,10,11,14,16,20,23,28,32,40,45,55,61,74,83,98,111,
%T 130,148,172,195,224,253,291,327,374,420,481,539,612,683,775,865,976,
%U 1087,1224,1365,1530,1701,1902,2113,2358,2613,2910,3221,3584,3960
%N McKay-Thompson series of class 54c for the Monster group.
%H Seiichi Manyama, <a href="/A112194/b112194.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%H William J. Keith, <a href="https://arxiv.org/abs/1911.04755">Partitions into parts simultaneously regular, distinct, and/or flat</a>, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of q^(1/6) * eta(q^3)^2 / (eta(q) * eta(q^9)) in powers of q. - _Michael Somos_, Aug 28 2015
%F Euler transform of period 9 sequence [ 1, 1, -1, 1, 1, -1, 1, 1, 0, ...]. - _Michael Somos_, Aug 28 2015
%F G.f. is a period 1 Fourier Series which satisifies f(-1 / (324 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 28 2015
%F G.f. is (1/q) f(q^18)^2/(f(q^6) f(q^54)) with f(q) the product of 1-q^k over all k; this is (1/q) times the generating function for partitions with parts not divisible by 3 which further must appear fewer than 3 times, magnified by the substitution q^6 for q. - _William J. Keith_, Jul 21 2015
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (6^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e G.f. = 1 + x + 2*x^2 + x^3 + 3*x^4 * 3*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + ...
%e T54c = 1/q + q^5 + 2*q^11 + q^17 + 3*q^23 + 3*q^29 + 4*q^35 + 4*q^41 +...
%t Series[Product[((1-q^(3k))^2)/((1-q^k)(1-q^(9k))), {k, 1, 55}], {q, 0, 55}] (* _William J. Keith_, Jul 21 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^9]) , {x, 0, n}]; (* _Michael Somos_, Aug 28 2015 *)
%o (PARI) N=66; q='q+O('q^N); Vec(prod(n=1,N,if(n%3,1+q^n+q^(2*n),1))) /* _Joerg Arndt_, Jul 22 2015 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A) * eta(x^9 + A)), n))}; /* _Michael Somos_, Aug 28 2015 */
%K nonn
%O 0,3
%A _Michael Somos_, Aug 28 2005