A139381 McKay-Thompson series of class 10E for the Monster group with a(0) = -3.

1, -3, 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

Offset

-1,2

Comments

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

Links

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

Michael 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 eta(q)^3 * eta(q^5) / eta(q^2) / eta(q^10)^3 in powers of q.

Expansion of q^(-1) * phi(-q) * f(-q) / (psi(q^5) * f(-q^10)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Euler transform of period 10 sequence [ -3, -2, -3, -2, -4, -2, -3, -2, -3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 20 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A095846.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + 4) * (20 + 6*v) - (v + 4) * (20 + v - u^2).

G.f.: (1 / x) * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) / ((1 - x^(2*k)) * (1 - x^(10*k))^3).

a(n) = A058101(n) = A132980(n) = A138516(n) unless n=0.

Convolution inverse of A095846.

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ -4 + (1/q) QPochhammer[ q^5, q^10]^5 QPochhammer[ -q, q], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ q]^3 QPochhammer[ q^5] / (QPochhammer[ q^2] QPochhammer[ q^10]^3), {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) / eta(x^2 + A) / eta(x^10 + A)^3, n))};

Crossrefs

Cf. A058101, A095846, A132980, A138516.

Sequence in context: A021888 A115310 A236172 * A225849 A038575 A347430

Adjacent sequences: A139378 A139379 A139380 * A139382 A139383 A139384

Keyword

sign

Author

Michael Somos, Apr 15 2008

Status

approved