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A096938 McKay-Thompson series of class 60F for the Monster group. +0
5
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 (list; graph; listen)
OFFSET

0,4

COMMENT

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.

REFERENCES

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

Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919(?).

LINKS

N. Chair, Partition identities from Partial Supersymmetry

FORMULA

Euler transform of period 10 sequence [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...]. - Vladeta Jovovic (vladeta(AT)Eunet.yu), 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.

EXAMPLE

a(8)=4, the number of partitions into distict 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 +...

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] (from Robert G. Wilson v Aug 19 2004)

PROGRAM

(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 */

CROSSREFS

Sequence in context: A029050 A066920 A035381 this_sequence A130084 A017981 A005863

Adjacent sequences: A096935 A096936 A096937 this_sequence A096939 A096940 A096941

KEYWORD

nonn

AUTHOR

Noureddine Chair (n.chair(AT)rocketmail.com), Aug 18 2004

EXTENSIONS

Definition corrected by Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 19 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 19 2004

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Last modified July 26 13:41 EDT 2008. Contains 142293 sequences.


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