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A186829
McKay-Thompson series of class 12A for the Monster group with a(0) = 6.
2
1, 6, 15, 32, 87, 192, 343, 672, 1290, 2176, 3705, 6336, 10214, 16320, 25905, 39936, 61227, 92928, 138160, 204576, 300756, 435328, 626727, 897408, 1271205, 1790592, 2508783, 3487424, 4824825, 6641664, 9083400, 12371904, 16778784, 22630912
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (1/q) * (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 12 sequence [ 6, -6, 12, 0, 6, -12, 6, 0, 12, -6, 6, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
Convolution inverse of A107653. Convolution square of A058571. Sixth convolution power of A112206.
G.f.: (1/x) * (Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(6*k - 3)))^6.
a(n) = -(-1)^n * A121666(n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 02 2015
Expansion of eta(q^2)^12 * eta(q^6)^12 / (eta(q)^6 * eta(q^3)^6 * eta(q^4)^6 * eta(q^12)^6) in powers of q. - G. A. Edgar, Mar 11 2017
EXAMPLE
G.f. = 1/q + 6 + 15*q + 32*q^2 + 87*q^3 + 192*q^4 + 343*q^5 + 672*q^6 + 1290*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^6, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^12 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^6, n))}
(PARI) q='q+O('q^50); Vec( eta(q^2)^12 * eta(q^6)^12 / (eta(q)^6 * eta(q^3)^6 * eta(q^4)^6 * eta(q^12)^6) ) \\ Joerg Arndt, Mar 11 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 27 2011
STATUS
approved