A133098 Expansion of q^(-1) * chi(-q)^2 * chi(-q^15)^2 / (chi(-q^3) * chi(-q^5)) in powers of q where chi() is a Ramanujan theta function.

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

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

McKay-Thompson series of class 30G for the Monster group with a(n) = -2. - Michael Somos, Oct 31 2015

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)^2 * eta(q^6) * eta(q^10) * eta(q^15)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^5) * eta(q^30)^2) in powers of q.

Euler transform of period 30 sequence [-2, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, ...].

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123630.

G.f.: (1/x) * Product_{k>0} (1 + x^(3*k)) * (1 + x^(5*k)) / ((1 + x^k)^2 * (1 + x^(15*k))^2 ).

a(n) = A058618(n) unless n=0. Convolution inverse of A123630.

a(n) = -(-1)^n * A145788(n). - Michael Somos, Oct 31 2015

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

a(2*n) = -A094023(n) = -A123630(n) if n>0. - Michael Somos, Oct 22 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/q (QPochhammer[ -q^3, q^3] QPochhammer[ -q^5, q^5]) / (QPochhammer[ -q, q] QPochhammer[ -q^15, q^15])^2, {q, 0, n}]; (* Michael Somos, Oct 22 2017 *)
QP = QPochhammer; a[n_] := SeriesCoefficient[(1/q)* (QP[q]^2*QP[q^6]* QP[q^10]*QP[q^15]^2)/(QP[q^2]^2*QP[q^3]*QP[q^5]*QP[q^30]^2), {q, 0, n}]; (* G. C. Greubel, Oct 21 2017 *)

Prog

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

Crossrefs

Cf. A058618, A094023, A123630, A145788.

Sequence in context: A161040 A180234 A131059 * A145788 A323861 A117592

Adjacent sequences: A133095 A133096 A133097 * A133099 A133100 A133101

Keyword

sign

Author

Michael Somos, Sep 10 2007

Status

approved