A132980 - OEIS

%I #12 Mar 30 2017 11:21:00

%S 1,1,1,2,2,-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,

%U -136,-57,-24,-214,-88,201,82,30,308,122,-288,-117,-48,-440,-180,410,168,74,624,262,-578,-238,-96,-874,-356,804

%N McKay-Thompson series of class 10E for the Monster group with a(0) = 1.

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

%H Seiichi Manyama, <a href="/A132980/b132980.txt">Table of n, a(n) for n = -1..10000</a>

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

%F Expansion of (eta(q^5) / eta(q^10))^5 / (eta(q) / eta(q^2)) in powers of q.

%F Euler transform of period 10 sequence [ 1, 0, 1, 0, -4, 0, 1, 0, 1, 0, ...].

%F G.f. A(q) satisfies A(q^2) = - A(q) * A(-q). - _Michael Somos_, Jul 05 2014

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

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

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132985.

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

%F a(n) = A058101(n) unless n=0. Convolution inverse is A095813.

%e G.f. = 1/q + 1 + 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[ (1/q) QPochhammer[ q^5, q^10]^5 / QPochhammer[ q, q^2], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x*O(x^n); polcoeff( eta(x^2 + A) / eta(x + A) * (eta(x^5 + A) / eta(x^10 + A))^5, n))};

%Y Cf. A058101, A095813, A132985.

%K sign

%O -1,4

%A _Michael Somos_, Sep 07 2007