A163229 - OEIS

%I #7 Jul 04 2018 08:58:29

%S 2,16,96,448,1858,6896,23776,76608,234432,684336,1921472,5206208,

%T 13679490,34941120,87036576,211822976,504784704,1179589728,2707337056,

%U 6109982400,13575320320,29721857904,64184237216,136816242816

%N A bisection of A162584.

%H G. C. Greubel, <a href="/A163229/b163229.txt">Table of n, a(n) for n = 1..1001</a>

%F Define series bisections B_0(q) and B_1(q) of A162584, then

%F 2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4),

%F the McKay-Thompson series of class 16B for the Monster group (A029839).

%e G.f.: B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...

%e Bisection B_0(q) of A162584 begins:

%e B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; nmax =250; a[n_]:= SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], 2*n + 1];  Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 03 2018 *)

%o (PARI) {a(n)=local(L=sum(m=1, 2*n+1, 2*sigma(m)*2^valuation(m, 2)*x^m/m)+O(x^(2*n+2))); polcoeff(exp(L), 2*n+1)}

%Y Cf. A162584, A163228 (B_0), A029839 (T16B).

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jul 26 2009