A226006 - OEIS

A226006

McKay-Thompson series of class 21B for the Monster group with a(0) = -1.

1, -1, -1, -1, 1, 2, -1, 3, -1, -1, -2, 0, 1, -2, 4, -1, -3, -4, 3, 3, -2, 10, -2, -6, -7, 3, 8, -6, 16, -4, -10, -12, 4, 9, -9, 24, -6, -14, -17, 8, 14, -12, 41, -9, -26, -30, 15, 30, -21, 64, -16, -35, -45, 16, 35, -33, 90, -21, -55, -66, 32, 54, -44, 140

(list; graph; refs; listen; history; text; internal format)

OFFSET

-1,6

LINKS

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Index entries for McKay-Thompson series for Monster simple group

FORMULA

Expansion of eta(q) * eta(q^3) / (eta(q^7) * eta(q^21)) in powers of q.

Euler transform of period 21 sequence [ -1, -1, -2, -1, -1, -2, 0, -1, -2, -1, -1, -2, -1, 0, -2, -1, -1, -2, -1, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (u*v + 7) - (u+v) * (u^2 - 3 * u*v + v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (21 t)) = 7 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226007.

G.f.: 1/x * Product_{k>0} (1 - x^k) * (1 - x^(3*k)) / ((1 - x^(7*k)) * (1 - x^(21*k))).

Convolution inverse is A226007.

a(n) = A058564(n) unless n=0.

EXAMPLE

G.f. = 1/q - 1 - q - q^2 + q^3 + 2*q^4 - q^5 + 3*q^6 - q^7 - q^8 - 2*q^9 + q^11 - ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] / (q QPochhammer[ q^7] QPochhammer[ q^21]), {q, 0, n}]; (* Michael Somos, Apr 12 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) / (eta(x^7 + A) * eta(x^21 + A)), n))};

CROSSREFS

Cf. A058564, A226007.

Sequence in context: A247564 A193870 A058564 * A210943 A260870 A282497

Adjacent sequences: A226003 A226004 A226005 * A226007 A226008 A226009

KEYWORD

sign

AUTHOR

Michael Somos, May 22 2013

STATUS

approved