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A187091 McKay-Thompson series of class 12H for the Monster group with a(0) = 4. 4
1, 4, 14, 36, 85, 180, 360, 684, 1246, 2196, 3754, 6264, 10226, 16380, 25804, 40032, 61275, 92628, 138452, 204804, 300040, 435672, 627356, 896400, 1271525, 1791324, 2507426, 3488472, 4825531, 6638688, 9085888, 12373992, 16772908, 22633812, 30411780, 40695048 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

COMMENTS

The reason for having multiple versions of these McKay-Thompson series is that each one is a quotient of Dedekind eta-functions, differing only in their constant terms. Each one is equally important.

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

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

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

Expansion of (eta(q^3) * eta(q^4) / (eta(q) * eta(q^12)))^4 in powers of q.

Euler transform of period 12 sequence [ 4, 4, 0, 0, 4, 0, 4, 0, 0, 4, 4, 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 square of A058578. a(n) = A058486(n) unless n=0.

a(n) = -(-1)^n * A193522(n). - Michael Somos, Sep 05 2015

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

EXAMPLE

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

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3] QPochhammer[ q^4] / (QPochhammer[ q] QPochhammer[ q^12]))^4, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] QPochhammer[ -q^2, q^2] QPochhammer[ q^3, q^6] QPochhammer[ q^6, q^12])^4, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

nmax = 50; CoefficientList[Series[Product[((1-x^(3*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(12*k))))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)))^4, n))};

CROSSREFS

Cf. A058486, A058578, A112192, A193522.

Sequence in context: A038164 A327382 A193522 * A034528 A128758 A258343

Adjacent sequences:  A187088 A187089 A187090 * A187092 A187093 A187094

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 06 2011

STATUS

approved

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Last modified September 15 16:31 EDT 2019. Contains 327078 sequences. (Running on oeis4.)