A162584 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.
1, 2, 8, 16, 50, 96, 240, 448, 1024, 1858, 3888, 6896, 13696, 23776, 44960, 76608, 139970, 234432, 414904, 684336, 1181568, 1921472, 3242928, 5206208, 8623104, 13679490, 22268752, 34941120, 56039936, 87036576, 137686048, 211822976
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 + ... 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 + ... The log of the g.f. of the Partition numbers (A000041) is: 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 + ... The log of the g.f. of the binary partitions (A000123) is: 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 + ... From _Paul D. Hanna_, Jul 26 2009: (Start) BISECTIONS begin: B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ... B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ... where 2*B_0(q)/B_1(q) = T16B(q): T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ... which is a g.f. of A029839. (End)
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Crossrefs
Programs
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Mathematica
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 *) 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 *) 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 *) -
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)}
Formula
From Paul D. Hanna, Jul 26 2009: (Start)
Define series BISECTIONS A(q) = B_0(q) + B_1(q), then
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)
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
Compare to the previous formula: 1/Product_{n>=0} Theta3(q^(2^n))^(2^n) = Theta4(q). - Joerg Arndt, Aug 03 2011
Comments