A045488 - OEIS

%I #27 Mar 12 2021 22:24:42

%S 1,1,6,4,-3,-12,-8,12,30,20,-30,-72,-46,60,156,96,-117,-300,-188,228,

%T 552,344,-420,-1008,-603,732,1770,1048,-1245,-2976,-1776,2088,4908,

%U 2900,-3420,-7992,-4658,5460,12756,7408,-8583,-19944,-11564,13344,30756,17744,-20448,-46944,-26916

%N McKay-Thompson series of class 6E for the Monster group with a(0) = 1.

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. C. Greubel, <a href="/A045488/b045488.txt">Table of n, a(n) for n = -1..1000</a>

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%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 J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

%H Morris Newman, <a href="http://plms.oxfordjournals.org/content/s3-9/3/373.extract">Construction and application of a class of modular functions. II</a>. Proc. London Math. Soc. (3) 9 1959 373-387.

%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 (1/q) * a(q^2) * psi(q) / psi(q^3)^3 in powers of q where psi() is a Ramanujan theta function and a() is a cubic AGM theta function. - _Michael Somos_, May 22 2015

%F Expansion of 6 + eta(q)^5 * eta(q^3) / (eta(q^2) * eta(q^6)^5) in powers of q. - _Michael Somos_, May 22 2015

%F a(n) = A007258(n) = A105559(n) = A128632(n) = A128633(n) = A258094(n) unless n=0. - _Michael Somos_, May 22 2015

%e G.f. = 1/q + 1 + 6*q + 4*q^2 - 3*q^3 - 12*q^4 - 8*q^5 + 12*q^6 + 30*q^7 + ...

%t a[ n_] := SeriesCoefficient[ 6 + QPochhammer[ q]^5 QPochhammer[ q^3] / (q QPochhammer[ q^2] QPochhammer[ q^6]^5), {q, 0, n}]; (* _Michael Somos_, May 22 2015 *)

%t a[ n_] := SeriesCoefficient[ -2 + (1/q) (QPochhammer[ q^2] QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^6]^3))^3, {q, 0, n}]; (* _Michael Somos_, May 22 2015 *)

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 3, 0, q^3] + 3 EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q]) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)]^3, {q, 0, n}]; (* _Michael Somos_, May 22 2015 *)

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( 6*x + eta(x + A)^5 * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)^5), n))}; /* _Michael Somos_, May 22 2015 */

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( -2*x + (eta(x^2 + A) * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^3))^3, n))}; /* _Michael Somos_, May 22 2015 */

%Y Cf. A007258, A105559, A128632, A128633, A258094.

%K sign

%O -1,3

%A _N. J. A. Sloane_