A145788 McKay-Thompson series of class 60C for the Monster group with a(0) = 2.

1, 2, 1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 9, 10, 11, 14, 18, 20, 22, 27, 32, 36, 40, 48, 57, 63, 70, 82, 95, 106, 119, 137, 158, 175, 195, 222, 252, 280, 311, 352, 397, 439, 486, 546, 611, 676, 747, 834, 929, 1024, 1128, 1253, 1389, 1528, 1679, 1857, 2052, 2250

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) * (chi(q) * chi(q^15))^2 / (chi(q^3) * chi(q^5)) in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2)^4 * eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20) * eta(q^30)^4) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^10) * eta(q^15) * eta(q^60))^2 in powers of q.

Euler transform of period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = f(t) where q = exp(2 Pi i t).

a(n) = A058727(n) = A145725(n) unless n=0. a(n) = -(-1)^n * A133098(n).

Convolution inverse of A145786.

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

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

Mathematica

a[ n_] := SeriesCoefficient[ 1/x (QPochhammer[ -x, x^2] QPochhammer[ -x^15, x^30])^2 / (QPochhammer[ -x^3, x^6] QPochhammer[ -x^5, x^10]), {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A) * eta(x^30 + A)^4) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^15 + A) * eta(x^60 + A))^2, n))};

Crossrefs

Cf. A058727, A133098, A145725, A145786.

Sequence in context: A180234 A131059 A133098 * A323861 A117592 A117942

Adjacent sequences: A145785 A145786 A145787 * A145789 A145790 A145791

Keyword

nonn

Author

Michael Somos, Oct 23 2008

Status

approved