A132976 McKay-Thompson series of class 36B for the Monster group with a(0) = -1.

1, -1, 0, -1, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 2, 0, 0, -3, 0, 0, 1, 0, 0, 4, 0, 0, -4, 0, 0, 1, 0, 0, 4, 0, 0, -6, 0, 0, 1, 0, 0, 5, 0, 0, -8, 0, 0, 1, 0, 0, 8, 0, 0, -10, 0, 0, 2, 0, 0, 11, 0, 0, -14, 0, 0, 4, 0, 0, 14, 0, 0, -19, 0, 0, 4, 0, 0, 17, 0, 0, -24, 0, 0, 4, 0, 0, 23

Offset

-1,22

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 eta(q) * eta(q^4) * eta(q^18) / (eta(q^2) * eta(q^9) * eta(q^36)) in powers of q.

Expansion of psi(-q) / (q * psi(-q^9)) = -1 + chi(q^9)^3 / (q * chi(q^3)) in powers of q where psi(), chi() are Ramanujan theta functions.

Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, -1, -1, 0, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u3 * u6 - (u1 + u2 + u1*u2) * (u3 + u6 + 3).

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132975.

a(3*n + 1) = 0. a(3*n) = 0 unless n=0. a(3*n - 1) = A062244(n). a(2*n) = -A139032(n). a(6*n - 1) = A132179(n). a(6*n + 2) = -A092848(n).

a(n) = -(-1)^n * A143840(n). Convolution inverse of A132975.

Example

G.f. = 1/q - 1 - q^2 + q^5 + q^8 - q^11 + q^17 - 2*q^20 + 2*q^26 - 3*q^29 + ...

Mathematica

a[ n_] := If[ n < -1, 0, SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(9/2)], {q, 0, n}]];
QP = QPochhammer; s = QP[q] * QP[q^4] * (QP[q^18] / (QP[q^2] * QP[q^9] * QP[q^36])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x*O(x^n) ; polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A) / (eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Crossrefs

Cf. A062244, A092848, A132975, A139032, A143840.

Sequence in context: A117896 A356735 A275962 * A143840 A028649 A305558

Adjacent sequences: A132973 A132974 A132975 * A132977 A132978 A132979

Keyword

sign

Author

Michael Somos, Sep 07 2007

Status

approved