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A161970 McKay-Thompson series of class 28C for the Monster group with a(0) = -1. 3
1, -1, -1, 0, 1, 0, -1, 0, 3, 0, -2, 0, 2, 0, -5, 0, 6, 0, -7, 0, 7, 0, -9, 0, 12, 0, -13, 0, 16, 0, -20, 0, 25, 0, -27, 0, 31, 0, -38, 0, 44, 0, -51, 0, 58, 0, -69, 0, 80, 0, -92, 0, 102, 0, -118, 0, 141, 0, -157, 0, 177, 0, -203, 0, 234, 0, -261, 0, 292, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,9

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^2) * chi(-q^7) * chi(-q^14) in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q) * eta(q^7) / (eta(q^4) * eta(q^28)) in powers of q.

Euler transform of period 28 sequence [ -1, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, -1, -2, -1, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, -1, 0, ...].

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123648.

a(n) = A230446(n) unless n=0. a(n) = -(-1)^n * A230446(n). a(2*n) = 0 unless n=0. a(2*n - 1) = A058608(n).

Convolution inverse is A123648.

EXAMPLE

G.f. = 1/q - 1 - q + q^3 - q^5 + 3*q^7 - 2*q^9 + 2*q^11 - 5*q^13 + 6*q^15 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q] QPochhammer[ q^7] / (QPochhammer[ q^4] QPochhammer[ q^28]), {q, 0, n}]; (* Michael Somos, Oct 18 2013 *)

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ q^2, q^4] QPochhammer[ q^7, q^14] QPochhammer[ q^14, q^28], {q, 0, n}]; (* Michael Somos, Sep 06 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x*O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^4 + A) * eta(x^28 + A)), n))};

CROSSREFS

Cf. A058608, A123648, A230446.

Sequence in context: A118514 A190544 A172293 * A230446 A260737 A059339

Adjacent sequences:  A161967 A161968 A161969 * A161971 A161972 A161973

KEYWORD

sign

AUTHOR

Michael Somos, Jun 22 2009

STATUS

approved

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Last modified September 23 11:45 EDT 2021. Contains 347612 sequences. (Running on oeis4.)