A132980 McKay-Thompson series of class 10E for the Monster group with a(0) = 1.

1, 1, 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

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

Formula

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

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

Euler transform of period 10 sequence [ 1, 0, 1, 0, -4, 0, 1, 0, 1, 0, ...].

G.f. A(q) satisfies A(q^2) = - A(q) * A(-q). - Michael Somos, Jul 05 2014

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

G.f. A(q) satisfies 0 = f(A(q), A(q^3)) where f(u, v) = (u - v)^4 - u * v * (u^2 - 3 * u - 4) * (v^2 - 3 * v - 4).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132985.

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

a(n) = A058101(n) unless n=0. Convolution inverse is A095813.

Example

G.f. = 1/q + 1 + 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[ (1/q) QPochhammer[ q^5, q^10]^5 / QPochhammer[ q, q^2], {q, 0, n}];

Prog

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

Crossrefs

Cf. A058101, A095813, A132985.

Sequence in context: A223903 A112159 A058101 * A106823 A160096 A029446

Adjacent sequences: A132977 A132978 A132979 * A132981 A132982 A132983

Keyword

sign

Author

Michael Somos, Sep 07 2007

Status

approved