A093067 - OEIS

%I #13 Feb 14 2018 03:38:40

%S 1,-2,-1,4,-3,-2,11,-6,-11,20,-15,-16,43,-24,-32,76,-48,-58,144,-84,

%T -97,238,-144,-172,398,-234,-279,636,-372,-428,1012,-582,-678,1564,

%U -906,-1028,2389,-1362,-1576,3560,-2046,-2320,5290,-2988,-3407,7700,-4371,-4928

%N McKay-Thompson series of class 24f for the Monster group with a(0) = -2.

%H G. C. Greubel, <a href="/A093067/b093067.txt">Table of n, a(n) for n = -1..1000</a>

%F Expansion of (eta(q) * eta(q^5) / (eta(q^3) * eta(q^15)))^2 in powers of q.

%F Euler transform of period 15 sequence [ -2, -2, 0, -2, -4, 0, -2, -2, 0, -4, -2, 0, -2, -2, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - 4*u*v * (u + v) - u^2*v^2 - 9*u*v.

%F a(n) = A058589(n) unless n=0.

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

%t a[ n_] := SeriesCoefficient[ q^(-1) (QPochhammer[ q] QPochhammer[ q^5] / (QPochhammer[q^3] QPochhammer[q^15]))^2, {q, 0, n}] (* _Michael Somos_, Oct 23 2013 *)

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

%o (PARI) {a(n) = local(A, u, v); if( n<-1, 0, A = 1/x; for(k=0, n, u = A + x * O(x^k); v = subst(u, x, x^2); A += x^k * polcoeff(u^3 + v^3 - 4*u*v*(u + v) - u^2*v^2 - 9*u*v, k-5) / 2); polcoeff(A, n))}

%Y Cf. A058509.

%K sign

%O -1,2

%A _Michael Somos_, Mar 17 2004

%E Edited for better readability and more coherence. - _Michael Somos_, Oct 23 2013