A103262 McKay-Thompson series of class 36g for the Monster group.
1, 2, 3, 4, 5, 8, 11, 16, 21, 26, 34, 44, 58, 74, 93, 116, 143, 178, 221, 272, 332, 402, 487, 588, 710, 854, 1021, 1216, 1444, 1714, 2031, 2400, 2826, 3318, 3888, 4552, 5322, 6208, 7224, 8388, 9726, 11264, 13028, 15044, 17339, 19952, 22930, 26324, 30186
Offset: 0
Keywords
Examples
E.g., a(5)=8 because we have 5,5*,41,41*,4*1,4*1*,22*1,22*1* with all parts prime to 3. The parts congruent to 1,5 mod(6) are 5, 5*, 11111, 11111*, 1111*1*, 111*1*1*, 11*1*1*1*, 1*1*1*1*1*. T36g = 1/q + 2*q^5 + 3*q^11 + 4*q^17 + 5*q^23 + 8*q^29 + 11*q^35 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
- D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
- Index entries for McKay-Thompson series for Monster simple group
Crossrefs
Cf. A003105.
Programs
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Maple
series(product((1+x^k)^2/(1+x^(3*k))^2,k=1..100),x=0,100);
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Mathematica
CoefficientList[ Series[ Product[(1 + x^k)^2/(1 + x^(3k))^2, {k, 60}], {x, 0, 50}], x] (* Robert G. Wilson v, Feb 22 2005 *) eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/6)(eta[q^2]eta[q^3]/(eta[q]eta[q^6]))^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 06 2018 *) -
PARI
{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)/eta(x+A)/eta(x^6+A))^2, n))} /* Michael Somos, Sep 10 2005 */
Formula
G.f.: product_{k>0}((1+x^k)/(1+x^(3k)))^2= 1/product_{k>0}((1-x^(6k-1))*(1-x^(6k-5)))^2.
Expansion of q^(1/6)(eta(q^2)eta(q^3)/(eta(q)eta(q^6)))^2 in powers of q.
Euler transform of period 6 sequence [2, 0, 0, 0, 2, 0, ...]. - Michael Somos, Sep 10 2005
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Sep 01 2015
Extensions
More terms from Robert G. Wilson v, Feb 22 2005
Comments