A153765 - OEIS

%I #20 Mar 19 2017 01:39:27

%S 1,4,8,22,42,70,155,246,421,722,1101,1730,2761,4062,6106,9040,13065,

%T 18806,27081,37950,53183,74290,102213,140048,191612,258426,348300,

%U 467484,622023,825016,1090957,1432290,1875930,2448610,3179136,4114996

%N McKay-Thompson series of class 15A for the Monster group with a(0) = 4.

%H G. C. Greubel, <a href="/A153765/b153765.txt">Table of n, a(n) for n = -1..1000</a>

%F Expansion of (A(q) + 3 / A(q))^2 in powers of q^2 where A(q) is g.f. for A058624.

%F Expansion of 1 + A(q) - 1 / A(q) in powers of q where A(q) is g.f. for A153084.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v - 12) * (u^2 + 7*u*v + v^2) - u*v * (u*v - 63).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = f(t) where q = exp(2 Pi i t).

%F a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Mar 18 2017

%e T15A = 1/q + 4 + 8*q + 22*q^2 + 42*q^3 + 70*q^4 + 155*q^5 + 246*q^6 + 421*q^7 + ...

%t QP = QPochhammer; A = QP[q]*(QP[q^5]/(QP[q^3]*QP[q^15])); s = (A + 3*(q/A))^2 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)

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

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

%Y A058508(n) = a(n) unless n = 0. Convolution square of A058625.

%Y Cf. A134783. - _R. J. Mathar_, Jan 07 2009

%K nonn

%O -1,2

%A _Michael Somos_, Jan 01 2009