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 A103262 McKay-Thompson series of class 36g for the Monster group. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of partitions of n into distinct parts prime to 3, with 2 types of each part. This is also the number of partitions of n into parts with 2 types congruent to 1 or 5 mod(6). 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). 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 EXAMPLE 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 + ... MAPLE series(product((1+x^k)^2/(1+x^(3*k))^2, k=1..100), x=0, 100); 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 *) PROG (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 */ CROSSREFS Cf. A003105. Sequence in context: A333347 A302592 A078762 * A135318 A210671 A189761 Adjacent sequences:  A103259 A103260 A103261 * A103263 A103264 A103265 KEYWORD nonn AUTHOR Noureddine Chair, Feb 21 2005 EXTENSIONS More terms from Robert G. Wilson v, Feb 22 2005 STATUS approved

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Last modified April 16 05:26 EDT 2021. Contains 343030 sequences. (Running on oeis4.)