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%I #30 Feb 07 2023 09:08:59
%S 1,2,8,16,50,96,240,448,1024,1858,3888,6896,13696,23776,44960,76608,
%T 139970,234432,414904,684336,1181568,1921472,3242928,5206208,8623104,
%U 13679490,22268752,34941120,56039936,87036576,137686048,211822976
%N G.f.: A(x) = exp( 2*Sum_{n>=1} sigma(n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.
%C Log of the g.f. A(x) is formed from the term-wise product of the log of the g.f.s of the partition numbers A000041 and the binary partitions A000123.
%H Vaclav Kotesovec, <a href="/A162584/b162584.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%F From _Paul D. Hanna_, Jul 26 2009: (Start)
%F Define series BISECTIONS A(q) = B_0(q) + B_1(q), then
%F 2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4), the McKay-Thompson series of class 16B for the Monster group (A029839). (End)
%F G.f.: 1/Product_{n>=0} Theta4(q^(2^n))^(2^n) = 1 / ( E(1)^2*E(2)^3*E(4)^6*E(8)^12* ... * E(2^n)^A042950(n) * ... ) where E(n) = Product_{k>=1} (1-q^(n*k)). - _Joerg Arndt_, Mar 20 2010
%F Compare to the previous formula: 1/Product_{n>=0} Theta3(q^(2^n))^(2^n) = Theta4(q). - _Joerg Arndt_, Aug 03 2011
%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 + ...
%e log(A(x))/2 = x + 6*x^2/2 + 4*x^3/3 + 28*x^4/4 + 6*x^5/5 + 24*x^6/6 + 8*x^7/7 + 120*x^8/8 + ... + sigma(n)*A006519(n)*x^n/n + ...
%e The log of the g.f. of the Partition numbers (A000041) is:
%e x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 + 12*x^6/6 + ... + sigma(n)*x^n/n + ...
%e The log of the g.f. of the binary partitions (A000123) is:
%e x + x^2/2 + x^3/3 + 4*x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 + ... + A006519(n)*x^n/n + ...
%e From _Paul D. Hanna_, Jul 26 2009: (Start)
%e BISECTIONS begin:
%e B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
%e B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
%e where 2*B_0(q)/B_1(q) = T16B(q):
%e T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
%e which is a g.f. of A029839. (End)
%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}], n]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 03 2018 *)
%t nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 20 2020 *)
%t nmax = 40; CoefficientList[Series[Product[1/EllipticTheta[4, 0, x^(2^k)]^(2^k), {k, 0, 1 + Log[2, nmax]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 07 2023 *)
%o (PARI) {a(n)=local(L=sum(m=1,n,2*sigma(m)*2^valuation(m,2)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}
%Y Cf. A000203, A006519, A000041, A000123.
%Y Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. - _Paul D. Hanna_, Jul 26 2009
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 06 2009
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