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A bisection of A162584.
3

%I #7 Jul 04 2018 08:58:20

%S 1,8,50,240,1024,3888,13696,44960,139970,414904,1181568,3242928,

%T 8623104,22268752,56039936,137686048,331039232,780029536,1804321074,

%U 4102056144,9177497600,20225408480,43948974720,94236510112,199549448704

%N A bisection of A162584.

%H G. C. Greubel, <a href="/A163228/b163228.txt">Table of n, a(n) for n = 0..1000</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_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...

%e Bisection B_1(q) of A162584 begins:

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

%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]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 03 2018 *)

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

%Y Cf. A162584, A163229 (B_1), A029839 (T16B).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 26 2009