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#30 by OEIS Server at Tue Feb 07 09:08:59 EST 2023
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Vaclav Kotesovec, <a href="/A162584/b162584_1.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
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#29 by Vaclav Kotesovec at Tue Feb 07 09:08:59 EST 2023
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Discussion
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| Tue Feb 07
| 09:08
| OEIS Server: Installed new b-file as b162584.txt. Old b-file is now b162584_1.txt.
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#28 by Vaclav Kotesovec at Tue Feb 07 09:08:27 EST 2023
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G. C. GreubelVaclav Kotesovec, <a href="/A162584/b162584_1.txt">Table of n, a(n) for n = 0..100010000</a>> (terms 0..1000 from G. C. Greubel)
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#27 by Vaclav Kotesovec at Tue Feb 07 09:04:54 EST 2023
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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 *)
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approved
editing
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#26 by Vaclav Kotesovec at Tue Oct 20 05:51:08 EDT 2020
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#25 by Vaclav Kotesovec at Tue Oct 20 05:47:33 EDT 2020
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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 *)
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approved
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#24 by Susanna Cuyler at Wed Jul 04 08:57:39 EDT 2018
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#23 by Jon E. Schoenfield at Wed Jul 04 03:58:13 EDT 2018
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#22 by Jon E. Schoenfield at Wed Jul 04 03:58:10 EDT 2018
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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)
the McKay-Thompson series of class 16B for the Monster group (A029839). (End)
G.f.: 1/prod(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) = prod(Product_{k>=1, } (1-q^(n*k) ). [)). - _Joerg Arndt, _, Mar 20 2010]
Compare to the previous formula: 1/prod(Product_{n>=0, } Theta3(q^(2^n))^(2^n) ) = ) = Theta4(q). - Joerg Arndt, Aug 03 2011
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| EXAMPLE
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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 +... + ...
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 +... + ...
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 +... + ...
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 +... + ...
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 +... + ...
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| CROSSREFS
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Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. [From . - _Paul D. Hanna, _, Jul 26 2009]
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proposed
editing
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#21 by Michel Marcus at Wed Jul 04 00:42:01 EDT 2018
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