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A138516 McKay-Thompson series of class 10E for the Monster group with a(0) = 2. 8
1, 2, 1, 2, 2, -2, -1, 0, -4, -2, 5, 2, 0, 8, 2, -8, -3, -2, -14, -6, 14, 6, 4, 24, 12, -24, -11, -4, -40, -16, 38, 16, 5, 62, 24, -60, -24, -10, -94, -40, 91, 38, 18, 144, 62, -136, -57, -24, -214, -88, 201, 82, 30, 308, 122, -288, -117, -48, -440, -180, 410, 168, 74, 624, 262, -578, -238, -96, -874, -356, 804 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of q^(-1) * (psi(q) / psi(q^5))^2 in powers of q where psi() is a Ramanujan theta function.

Expansion of ((eta(q^2) / eta(q^10))^2 * eta(q^5) / eta(q))^2 in powers of q.

Euler transform of period 10 sequence [ 2, -2, 2, -2, 0, -2, 2, -2, 2, 0, ...].

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

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u - 5) * v * (v - 1) * (v - 5).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5 g(t) where q = exp(2 Pi i t) and g() is g.f. for A138517.

G.f.: (1/x) * (Product_{k>0} P(5,x^k) * P(10,x^k)^2)^(-2) where P(n,x) is the n-th cyclotomic polynomial.

EXAMPLE

1/q + 2 + q + 2*q^2 + 2*q^3 - 2*q^4 - q^5 - 4*q^7 - 2*q^8 + 5*q^9 + ...

MATHEMATICA

QP = QPochhammer; s = ((QP[q^2]/QP[q^10])^2*(QP[q^5]/QP[q]))^2 + O[q]^80; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 14 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( (eta(x^2 + A) / eta(x^10 + A))^2 * eta(x^5 + A) / eta(x + A))^2, n))}

CROSSREFS

A058101(n) = a(n) unless n=0. Convolution inverse of A138519. Convolution square of A138532.

Cf. A132980. [From R. J. Mathar, Dec 13 2008]

Sequence in context: A104605 A300953 A145740 * A180580 A026513 A106028

Adjacent sequences:  A138513 A138514 A138515 * A138517 A138518 A138519

KEYWORD

sign

AUTHOR

Michael Somos, Mar 23 2008

STATUS

approved

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Last modified January 20 17:05 EST 2019. Contains 319335 sequences. (Running on oeis4.)