A225849 - OEIS

%I #17 Mar 12 2021 22:24:46

%S 1,3,1,-2,2,2,-1,0,-4,2,5,-2,0,-8,2,8,-3,2,-14,6,14,-6,4,-24,12,24,

%T -11,4,-40,16,38,-16,5,-62,24,60,-24,10,-94,40,91,-38,18,-144,62,136,

%U -57,24,-214,88,201,-82,30,-308,122,288,-117,48,-440,180,410

%N McKay-Thompson series of class 20C for the Monster group with a(0) = 3.

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

%H G. C. Greubel, <a href="/A225849/b225849.txt">Table of n, a(n) for n = -1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(-1) * f(q)^3 / (f(-q^2) * f(-q^5)* f(-q^20)) in powers of q where f() is a Ramanujan theta function.

%F Expansion of q^(-1) * f(q, q)^2 / (f(q, q^9) * f(q^3, q^7)) in powers of q where f(, ) is Ramanujan's general theta functions.

%F Expansion of q^(-1) * (phi(q) / phi(q^5))^2 * chi(q^5)^5 / chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

%F Expansion of eta(q^2)^8 / ( eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.

%F Euler transform of period 20 sequence [ 3, -5, 3, -2, 4, -5, 3, -2, 3, -4, 3, -2, 3, -5, 4, -2, 3, -5, 3, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 4) * (u - 5) * v * (v - 4) * (v - 5).

%F G.f.: (1/x) * Product_{k>0} (1 - (-x)^k)^3 / ((1 - x^(2*k)) * (1 - x^(5*k)) * (1 - x^(20*k))).

%F a(n) = A112159(n) = A145740(n) = A223903(n) unless n=0. a(n) = -(-1)^n * A139381(n).

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

%t a[ n_] := SeriesCoefficient[ 4 + (1/q) QPochhammer[ q^5, q^10]^5 / QPochhammer[ q, q^2], {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ -q]^3 / (QPochhammer[ q^2] QPochhammer[ q^5] QPochhammer[ q^20]), {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 / ( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};

%Y Cf. A112159, A145740, A139381, A138516, A132980, A058101, A223903.

%K sign

%O -1,2

%A _Michael Somos_, May 17 2013