A215412 - OEIS

%I #36 Mar 12 2021 22:24:46

%S 1,-2,3,-2,3,-6,10,-12,15,-22,30,-36,44,-60,78,-96,117,-150,190,-228,

%T 276,-340,420,-504,603,-732,885,-1052,1245,-1488,1770,-2088,2454,

%U -2902,3420,-3996,4666,-5460,6378,-7400,8583,-9972,11566,-13344,15378,-17752,20448

%N McKay-Thompson series of class 18C for the Monster group with a(0) = -2.

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

%C A058533, A123676, A215412, A058644, A215413 are all essentially the same sequence. - _N. J. A. Sloane_, Aug 09 2012

%H G. C. Greubel, <a href="/A215412/b215412.txt">Table of n, a(n) for n = -1..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>, Commun. Algebra 22, No. 13, 5175-5193 (1994). See Table 4 18C.

%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>

%F Expansion of -3 + psi(q) / (q * psi(q^9)) + 3 * q * psi(q^9) / psi(q) in powers of q where psi() is a Ramanujan  theta function.

%F Expansion of (1/q) * (psi(q^3)^2 / (psi(q) * psi(q^9)))^2 in powers of q where psi() is a Ramanujan theta function.

%F Expansion of 3 * b(q) * c(q) * (b(q^6)^2 / (b(q^2) * c(q^2) * b(q^3)))^2 in powers of q where b(), c() are cubic AGM theta functions.

%F Expansion of (eta(q) * eta(q^6)^4 * eta(q^9))^2 / (eta(q^2) * eta(q^3) * eta(q^18))^4 in powers of q.

%F Euler transform of period 18 sequence [ -2, 2, 2, 2, -2, -2, -2, 2, 0, 2, -2, -2, -2, 2, 2, 2, -2, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 1) * (v - u^2) - 4 * v * (u - 1).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227587. - _Michael Somos_, Jul 16 2013

%F a(n) = A058533(n) = A123676(n) = A215413(n) unless n=0.

%F a(n) = -(-1)^n * A227585(n). - _Michael Somos_, Jul 16 2013

%F Convolution square of A112176. - _Michael Somos_, Jul 16 2013

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017

%e 1/q - 2 + 3*q - 2*q^2 + 3*q^3 - 6*q^4 + 10*q^5 - 12*q^6 + 15*q^7 - 22*q^8 + ...

%t QP = QPochhammer; s = (QP[q] * QP[q^6]^4 * QP[q^9])^2 / (QP[q^2] * QP[q^3] * QP[q^18])^4 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^4 * eta(x^9 + A))^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^18 + A))^4, n))}

%Y Cf. A058533, A112176, A123676, A215413, A227585, A227587.

%K sign

%O -1,2

%A _Michael Somos_, Aug 09 2012