%I #37 Mar 12 2021 22:24:42
%S 1,1,3,4,6,7,13,16,22,29,40,50,69,83,110,136,174,214,272,332,413,502,
%T 618,748,915,1095,1329,1590,1910,2272,2718,3216,3823,4508,5332,6262,
%U 7378,8630,10119,11802,13784,16023,18650,21612,25070,28972,33502,38610
%N McKay-Thompson series of class 34a for the Monster group.
%C Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H Vaclav Kotesovec, <a href="/A058639/b058639.txt">Table of n, a(n) for n = 0..10000</a>
%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 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of (psi(q^2) * phi(q^17) - q^4 * phi(q) * psi(q^34)) / (f(-q) * f(-q^17)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Dec 11 2008
%F Expansion of (F(q) - q^4 / F(q)) / (chi(-q) * chi(-q^17))^2 in powers of q where F(q) = G(q^17) / G(q), G(q) = chi(q) * chi(-q^2) and chi() is a Ramanujan theta function. - _Michael Somos_, Dec 11 2008
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (68 t)) = f(t) where q = exp(2 Pi i t). - _Michael Somos_, Dec 11 2008
%F a(n) ~ exp(2*Pi*sqrt(2*n/17)) / (2^(3/4) * 17^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018
%e T34a = 1/q + q + 3*q^3 + 4*q^5 + 6*q^7 + 7*q^9 + 13*q^11 + 16*q^13 + ...
%t QP = QPochhammer; s = QP[q^4]^2*(QP[q^34]^5/(QP[q]*QP[q^2]*QP[q^17]^3* QP[q^68]^2))-q^4*QP[q^2]^5*(QP[q^68]^2/(QP[q]^3*QP[q^4]^2*QP[q^17]* QP[q^34]))+O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 13 2015, adapted from PARI *)
%t nmax = 60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 * (1+x^(17*k)) * (1+x^(34*k-17))^2, {k, 1, nmax}] - x^4*Product[(1+x^k) * (1+x^(2*k-1))^2 * (1+x^(17*k)) * (1+x^(34*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 01 2017 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x + A) * eta(x^2 + A) * eta(x^17 + A)^3 * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^68 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^17 + A) * eta(x^34 + A)), n))} /* _Michael Somos_, Dec 15 2008 */
%Y Cf. A000521, A007240, A007241, A007267, A014708, A045478.
%Y Convolution square is A152944.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Nov 27 2000
%E Erroneous zero at start of sequence removed by _Michael Somos_, Sep 30 2009