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A128519 McKay-Thompson series of class 78B for the Monster group with a(0) = -1. 3
1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 2, -1, 0, 0, 1, -2, 2, -2, 1, 0, 1, -3, 4, -3, 2, -1, 2, -4, 5, -5, 3, -2, 3, -6, 8, -7, 4, -2, 5, -9, 11, -10, 6, -4, 6, -12, 16, -14, 8, -6, 11, -17, 21, -19, 13, -10, 14, -24, 30, -26, 17, -14, 21, -31, 38, -35, 25, -20, 26, -42, 52, -46, 33, -28, 38, -56, 68, -62, 47, -38, 49, -75 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,13

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1) * chi(-q) * chi(-q^39) / (chi(-q^3) * chi(-q^13)) in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q) * eta(q^6) * eta(q^26) * eta(q^39) / (eta(q^2) * eta(q^3) * eta(q^13) * eta(q^78)) in powers of q.

Euler transform of a period 78 sequence.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v - u^2) * (w^2 - v) - 2*u*w * (1 + v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (78 t)) = 1 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A262950.

G.f.: (1/x) * (Product_{k>0} P(x^k))^-1 where P(x) is the 78th cyclotomic polynomial of degree 24.

a(n) = A058755(n) unless n = 0.

Convolution inverse is A262950.

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/39)) / (2 * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018

EXAMPLE

G.f. = 1/q - 1 - q^4 + q^5 - q^6 + q^7 - q^10 + 2*q^11 - q^12 + q^15 + ...

MATHEMATICA

QP = QPochhammer; s = QP[q]*QP[q^6]*QP[q^26]*(QP[q^39]/(QP[q^2]*QP[q^3]* QP[q^13]*QP[q^78])) + O[q]^90; CoefficientList[s, q] (* Jean-Fran├žois Alcover, Nov 15 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^6 + A) * eta(x^26 + A) * eta(x^39 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^13 + A) * eta(x^78 + A)), n))};

CROSSREFS

Cf. A058755, A262950.

Sequence in context: A048571 A025880 A058755 * A303979 A301573 A061670

Adjacent sequences:  A128516 A128517 A128518 * A128520 A128521 A128522

KEYWORD

sign

AUTHOR

Michael Somos, Mar 06 2007

STATUS

approved

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Last modified June 2 10:43 EDT 2020. Contains 334770 sequences. (Running on oeis4.)