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Search: id:A096938
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| A096938 |
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McKay-Thompson series of class 60F for the Monster group. |
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5
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| 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)
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OFFSET
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0,4
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COMMENT
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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.
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REFERENCES
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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(?).
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LINKS
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N. Chair, Partition identities from Partial Supersymmetry
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FORMULA
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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.
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EXAMPLE
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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 +...
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MAPLE
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series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)), k+1..150), x=0, 100);
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MATHEMATICA
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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)
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PROGRAM
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(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 */
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CROSSREFS
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Sequence in context: A029050 A066920 A035381 this_sequence A130084 A017981 A005863
Adjacent sequences: A096935 A096936 A096937 this_sequence A096939 A096940 A096941
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KEYWORD
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nonn
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AUTHOR
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Noureddine Chair (n.chair(AT)rocketmail.com), Aug 18 2004
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EXTENSIONS
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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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