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A187045 McKay-Thompson series of class 12c for the Monster group with a(0) = 4. 3
1, 4, 5, 0, -5, 0, 9, 0, -14, 0, 19, 0, -34, 0, 55, 0, -69, 0, 104, 0, -164, 0, 209, 0, -283, 0, 413, 0, -539, 0, 712, 0, -968, 0, 1248, 0, -1642, 0, 2167, 0, -2731, 0, 3526, 0, -4592, 0, 5736, 0, -7244, 0, 9255, 0, -11520, 0, 14378, 0, -18018, 0, 22238, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

COMMENTS

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

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1001 from G. A. Edgar)

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.

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).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Expansion of (1/q) * chi(q)^5 * chi(-q) * chi(q^3) * chi(-q^3)^5 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2)^9 * eta(q^3)^4) / (eta(q)^4 * eta(q^4)^5 * eta(q^6)^3 * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 4, -5, 0, 0, 4, -6, 4, 0, 0, -5, 4, 0, ...].

a(2*n) = 0 unless n=0. a(n) = A186930(n) unless n=0. a(2*n - 1) = A058491(n).

EXAMPLE

G.f. = 1/q + 4 + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ -q^3, q^6] (QPochhammer[ -q, q^2] QPochhammer[ q^3, q^6])^5, {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 * eta(x^3 + A)^4) / (eta(x + A)^4 * eta(x^4 + A)^5 * eta(x^6 + A)^3 * eta(x^12 + A)), n))};

CROSSREFS

Cf. A058491, A186930.

Sequence in context: A213440 A011286 A246927 * A186930 A159567 A164357

Adjacent sequences:  A187042 A187043 A187044 * A187046 A187047 A187048

KEYWORD

sign

AUTHOR

Michael Somos, Mar 07 2011

STATUS

approved

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Last modified November 28 16:42 EST 2021. Contains 349413 sequences. (Running on oeis4.)