A058559 - OEIS

%I #24 Nov 15 2020 12:58:47

%S 1,0,3,-4,4,-4,7,-12,13,-16,22,-28,38,-44,55,-72,83,-104,129,-156,187,

%T -220,273,-328,384,-452,539,-652,757,-880,1041,-1220,1428,-1652,1924,

%U -2244,2585,-2992,3458,-3992,4581,-5244,6053,-6936,7910,-9024,10303,-11784,13380,-15176,17257,-19584

%N McKay-Thompson series of class 20d for Monster.

%H G. C. Greubel, <a href="/A058559/b058559.txt">Table of n, a(n) for n = -1..5000</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. Algebra 22, No. 13, 5175-5193 (1994).

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of sqrt( 4 + (eta(q)*eta(q^5)/(eta(q^2)*eta(q^10)))^4 ) in powers of q. - _G. C. Greubel_, Jun 14 2018

%F a(n) ~ -(-1)^n * exp(sqrt(2*n/5)*Pi) / (2^(5/4) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T20d = 1/q + 3*q^3 - 4*q^5 + 4*q^7 - 4*q^9 + 7*q^11 - 12*q^13 + 13*q^15 - ...

%t QP = QPochhammer; A = (QP[q]*(QP[q^5]/QP[q^2]/QP[q^10]))^4 + 4*q + O[q]^40;

%t CoefficientList[Sqrt[A], q] (* _Georg Fischer_, Nov 15 2020 *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff(sqrtn( (eta(x + A) * eta(x^5 + A) / eta(x^2 + A) / eta(x^10 + A))^4 + 4 * x,2), n))} /* _Georg Fischer_, Nov 15 2020 [adapted from Michael Somos' PARI in A058098] */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%Y Cf. A058098, A112158, A132040.

%K sign

%O -1,3

%A _N. J. A. Sloane_, Nov 27 2000

%E Terms a(12) onward added by _G. C. Greubel_, Jun 14 2018