A187198 - OEIS

%I #22 Jun 20 2018 03:20:43

%S 1,5,14,36,85,180,360,684,1246,2196,3754,6264,10226,16380,25804,40032,

%T 61275,92628,138452,204804,300040,435672,627356,896400,1271525,

%U 1791324,2507426,3488472,4825531,6638688,9085888,12373992,16772908,22633812

%N McKay-Thompson series of class 12H for the Monster group with a(0) = 5.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A187198/b187198.txt">Table of n, a(n) for n = -1..2500</a>

%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (b(q^2) * c(q^2))^3 / (b(q)^2 * c(q) * b(q^4) * c(q^4)^2) in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of (1/q) * chi(q) * chi(q^3) * chi(-q^6)^4 / chi(-q)^4 in powers of q where chi() is a Ramanujan theta function.

%F Expansion of (eta(q^2) * eta(q^6))^6 / (eta(q)^5 * eta(q^3) * eta(q^4) * eta(q^12)^5) in powers of q.

%F Euler transform of period 12 sequence [ 5, -1, 6, 0, 5, -6, 5, 0, 6, -1, 5, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) = A058486(n) = A187091(n) unless n=0.

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e 1/q + 5 + 14*q + 36*q^2 + 85*q^3 + 180*q^4 + 360*q^5 + 684*q^6 + 1246*q^7 + ...

%t QP = QPochhammer; s = (QP[q^2]*QP[q^6])^6/(QP[q]^5*QP[q^3]*QP[q^4]* QP[q^12]^5) + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^6 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)^5), n))}

%Y Cf. A058486, A187091.

%K nonn

%O -1,2

%A _Michael Somos_, Mar 06 2011