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

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, 92, 104, 118, 135, 152, 171, 194, 218, 244, 275, 308, 344, 386, 432, 481, 537, 598, 664, 738, 819, 908, 1006, 1114, 1232, 1362, 1503, 1658, 1828, 2012, 2214, 2436, 2676

Offset: 0

Author

Noureddine Chair, Aug 18 2004

Keywords

• nonn

Comments

The inverted graded parafermionic partition function.

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.

Number of partitions of n into distinct parts in which no part is a multiple of 5.

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.

Examples

			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.
T60F = 1/q + q^5 + q^11 + 2*q^17 + 2*q^23 + 2*q^29 + 3*q^35 + 4*q^41 +...
		

References

• T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Cristina Ballantine and Brooke Feigon, Truncated Theta Series Related to the Jacobi Triple Product Identity, arXiv:2401.04019 [math.CO], 2024. See page 16.
• Nayandeep Deka Baruah and Abhishek Sarma, Arithmetic properties of 5-regular partitions into distinct parts, arXiv:2411.02978 [math.NT], 2024. See p. 2.
• N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
• Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
• Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.
• Index entries for sequences related to groups
• Index entries for McKay-Thompson series for Monster simple group

Crossrefs

Cf. A133563.

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).

Programs

Maple

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)),k+1..150),x=0,100);

Mathematica

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 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
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 *)

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 */

Formula

Euler transform of period 10 sequence [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...]. - Vladeta Jovovic, Aug 19 2004

Expansion of q^(1/6)eta(q^2)eta(q^5)/(eta(q)eta(q^10)) in powers of q.

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

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.

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

Extensions

Definition corrected by Vladeta Jovovic, Aug 19 2004

More terms from Robert G. Wilson v, Aug 19 2004