A132040 McKay-Thompson series of class 10B for the Monster group with a(0) = -4.

1, -4, 6, -8, 17, -32, 54, -80, 116, -192, 290, -408, 585, -832, 1192, -1648, 2237, -3072, 4156, -5576, 7414, -9824, 12964, -16896, 22002, -28544, 36794, -47184, 60185, -76736, 97388, -122864, 154615, -194048, 242904, -302800, 376271, -466720, 577176, -711840

Offset

-1,2

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^-1 * (chi(-q) * chi(-q^5))^4 in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q) * eta(q^5) / (eta(q^2) * eta(q^10)))^4 in powers of q.

Euler transform of period 10 sequence [ -4, 0, -4, 0, -8, 0, -4, 0, -4, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v*(u^2 - v) + 8*u * (v + 2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 16 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093861.

G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(5*k)))^-4.

a(n) = A058098(n) unless n = 0. a(n) = -(-1)^n * A112158(n) unless n = 0.

Convolution inverse is A093831. - Michael Somos, Apr 26 2015

a(n) = -(-1)^n * A210459(n). - Michael Somos, Nov 01 2015, corrected by Vaclav Kotesovec, Sep 08 2017

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

G.f. = 1/q - 4 + 6*q - 8*q^2 + 17*q^3 - 32*q^4 + 54*q^5 - 80*q^6 + 116*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] QPochhammer[ -q^5, q^5])^-4, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A) / (eta(x^2 + A) * eta(x^10 + A)))^4, n))};

Crossrefs

Cf. A058098, A093831, A112158, A210459.

Sequence in context: A185292 A022599 A112160 * A210459 A346868 A279896

Adjacent sequences: A132037 A132038 A132039 * A132041 A132042 A132043

Keyword

sign

Author

Michael Somos, Aug 07 2007

Status

approved