A094362 - OEIS

%I #17 Jun 16 2018 13:41:01

%S 1,1,2,2,4,5,7,9,13,16,22,27,36,43,56,68,87,104,130,156,193,230,281,

%T 333,404,477,572,673,802,940,1113,1299,1531,1780,2085,2418,2820,3259,

%U 3784,4362,5047,5799,6685,7662,8806,10066,11532,13152,15026,17098,19482

%N McKay-Thompson series of class 39C for the Monster group with a(0) = 1.

%H G. C. Greubel, <a href="/A094362/b094362.txt">Table of n, a(n) for n = -1..1000</a>

%F Expansion of (eta(q^3) * eta(q^13)) / (eta(q) * eta(q^39)) in powers of q.

%F Euler transform of period 39 sequence [ 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 + 2*u*v*(u + v) - u^2*v^2 - u*v.

%F G.f.: x^-1 * Product_{k>0} (1 - x^(3*k)) * (1 - x^(13*k)) / ((1 - x^k) * (1 - x^(39*k))).

%F a(n) = A058661(n) unless n=0. Convolution inverse of A094363.

%F a(n) ~ exp(4*Pi*sqrt(n/39)) / (sqrt(2) * 39^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015

%e 1/q + 1 + 2*q + 2*q^2 + 4*q^3 + 5*q^4 + 7*q^5 + 9*q^6 + 13*q^7 + 16*q^8 + 22*q^9 + ...

%t QP = QPochhammer; s = QP[q^3]*(QP[q^13]/(QP[q]*QP[q^39])) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^13 + A) / (eta(x + A) * eta(x^39 + A)), 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 + 2*u*v*(u + v) - u^2*v^2 - u*v, k-5) / 2); polcoeff(A, n))}

%Y Cf. A058661. [From _R. J. Mathar_, Dec 15 2008]

%Y Cf. A094363.

%K nonn

%O -1,3

%A _Michael Somos_, May 03 2004