cp's OEIS Frontend

A096938 McKay-Thompson series of class 60F for the Monster group.

%I A096938 #49 Nov 11 2024 22:29:57
%S A096938 1,1,1,2,2,2,3,4,4,6,7,8,10,12,14,16,19,22,26,30,35,41,47,54,62,70,80,
%T A096938 92,104,118,135,152,171,194,218,244,275,308,344,386,432,481,537,598,
%U A096938 664,738,819,908,1006,1114,1232,1362,1503,1658,1828,2012,2214,2436,2676
%N A096938 McKay-Thompson series of class 60F for the Monster group.
%C A096938 The inverted graded parafermionic partition function.
%C A096938 Also number of partitions of n into parts congruent to {1,3,7,9} mod 10. Also number of partitions of n into odd parts parts in which no part appears more than 4 times.
%C A096938 Number of partitions of n into distinct parts in which no part is a multiple of 5.
%C A096938 This generating function is a generalization of the sequences A003105 and A006950. It arose in my recent work on partial supersymmetry in writing the graded parafermionic partition function in which I obtained a more general formula.
%D A096938 T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976
%H A096938 Seiichi Manyama, <a href="/A096938/b096938.txt">Table of n, a(n) for n = 0..1000</a>
%H A096938 Cristina Ballantine and Brooke Feigon, <a href="https://arxiv.org/abs/2401.04019">Truncated Theta Series Related to the Jacobi Triple Product Identity</a>, arXiv:2401.04019 [math.CO], 2024. See page 16.
%H A096938 Nayandeep Deka Baruah and Abhishek Sarma, <a href="https://arxiv.org/abs/2411.02978">Arithmetic properties of 5-regular partitions into distinct parts</a>, arXiv:2411.02978 [math.NT], 2024. See p. 2.
%H A096938 N. Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H A096938 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
%H A096938 Donald Spector, <a href="http://arxiv.org/abs/hep-th/9710002">Duality, partial supersymmetry and arithmetic number theory</a>, arXiv:hep-th/9710002, 1997.
%H A096938 Donald Spector, <a href="http://dx.doi.org/10.1063/1.532269">Duality, partial supersymmetry and arithmetic number theory</a>, J. Math. Phys. Vol. 39, 1998, p. 1919.
%H A096938 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A096938 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A096938 Euler transform of period 10 sequence [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...]. - _Vladeta Jovovic_, Aug 19 2004
%F A096938 Expansion of q^(1/6)eta(q^2)eta(q^5)/(eta(q)eta(q^10)) in powers of q.
%F A096938 Given g.f. A(x), then B(x)=(A(x^6)/x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(u^3+v^3)(1+uv)-uv(1-uv)^2. - _Michael Somos_, Jan 18 2005
%F A096938 G.f.: 1/product_{k>=1} (1-x^k+x^(2*k)-x^(3*k)+x^(4*k)) = 1/Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
%F A096938 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)) * (1 - (3*sqrt(15)/(16*Pi) + Pi/(6*sqrt(15))) / sqrt(n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 21 2017
%e A096938 a(8)=4, the number of partitions into distinct parts that exclude the number 5 because we can write 8=7+1=6+2=4+3+1.
%e A096938 T60F = 1/q + q^5 + q^11 + 2*q^17 + 2*q^23 + 2*q^29 + 3*q^35 + 4*q^41 +...
%p A096938 series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)),k+1..150),x=0,100);
%t A096938 CoefficientList[ Series[ Product[1/(1 - x^k + x^(2k) - x^(3k) + x^(4k)), {k, 70}], {x, 0, 60}], x] (* _Robert G. Wilson v_, Aug 19 2004 *)
%t A096938 nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)
%t A096938 QP = QPochhammer; s = QP[q^2]*(QP[q^5]/(QP[q]*QP[q^10])) + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015 *)
%o A096938 (PARI) {a(n)=local(A); if(n<0,0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^5+A)/eta(x+A)/eta(x^10+A), n))} /* _Michael Somos_, Jan 18 2005 */
%Y A096938 Cf. A133563.
%Y A096938 Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).
%K A096938 nonn
%O A096938 0,4
%A A096938 _Noureddine Chair_, Aug 18 2004
%E A096938 Definition corrected by _Vladeta Jovovic_, Aug 19 2004
%E A096938 More terms from _Robert G. Wilson v_, Aug 19 2004