A225849 McKay-Thompson series of class 20C 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

Formula

Expansion of q^(-1) * f(q)^3 / (f(-q^2) * f(-q^5)* f(-q^20)) in powers of q where f() is a Ramanujan theta function.

Expansion of q^(-1) * f(q, q)^2 / (f(q, q^9) * f(q^3, q^7)) in powers of q where f(, ) is Ramanujan's general theta functions.

Expansion of q^(-1) * (phi(q) / phi(q^5))^2 * chi(q^5)^5 / chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

Expansion of eta(q^2)^8 / ( eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [ 3, -5, 3, -2, 4, -5, 3, -2, 3, -4, 3, -2, 3, -5, 4, -2, 3, -5, 3, 0, ...].

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

G.f.: (1/x) * Product_{k>0} (1 - (-x)^k)^3 / ((1 - x^(2*k)) * (1 - x^(5*k)) * (1 - x^(20*k))).

a(n) = A112159(n) = A145740(n) = A223903(n) unless n=0. a(n) = -(-1)^n * A139381(n).

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^2], {q, 0, n}];
a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ -q]^3 / (QPochhammer[ q^2] QPochhammer[ q^5] QPochhammer[ q^20]), {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / ( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A112159, A145740, A139381, A138516, A132980, A058101, A223903.

Sequence in context: A115310 A236172 A139381 * A038575 A347430 A346491

Adjacent sequences: A225846 A225847 A225848 * A225850 A225851 A225852

Keyword

sign

Author

Michael Somos, May 17 2013

Status

approved