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A132321 McKay-Thompson series of class 30C for the Monster group with a(0) = -1. 4
1, -1, 0, -2, 2, -2, 3, -2, 5, -6, 5, -6, 9, -10, 10, -16, 17, -18, 25, -26, 31, -38, 37, -48, 60, -62, 68, -84, 95, -104, 125, -134, 154, -182, 192, -220, 257, -274, 309, -360, 394, -434, 492, -544, 607, -688, 740, -824, 944, -1018, 1123, -1266, 1377, -1524 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,4

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = -1..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q) * eta(q^3) * eta(q^5) * eta(q^15) / (eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30)) in powers of q.

Euler transform of period 30 sequence [ -1, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1, 0, -1, 0, -4, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -1, 0, ...].

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123632.

G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) * (1 + x^(5*k)) * (1 + x^(15*k)))^-1.

Expansion of q^-1 * chi(-q) * chi(-q^3) * chi(-q^5) * chi(-q^15) in powers of q where chi() is a Ramanujan theta function.

a(n) = A058614(n) unless n = 0.

Convolution inverse of A123632.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/15)) / (2*15^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

EXAMPLE

G.f. = 1/q - 1 - 2*q^2 + 2*q^3 - 2*q^4 + 3*q^5 - 2*q^6 + 5*q^7 - 6*q^8 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10] QPochhammer[ q^15, q^30] / q, {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)

PROG

(PARI) {a(n) = my(A); if(n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A) / (eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A)), n))};

CROSSREFS

Cf. A058614, A123632.

Sequence in context: A319478 A104239 A283751 * A058726 A058614 A122765

Adjacent sequences:  A132318 A132319 A132320 * A132322 A132323 A132324

KEYWORD

sign

AUTHOR

Michael Somos, Aug 18 2007

STATUS

approved

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)