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Search: id:A073252
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| A073252 |
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Expansion of Product (1+q^(2*m-1))^2, m=1..infinity. |
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+0
3
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| 1, 2, 1, 2, 4, 4, 5, 6, 9, 12, 13, 16, 21, 26, 29, 36, 46, 54, 62, 74, 90, 106, 122, 142, 171, 200, 227, 264, 311, 358, 408, 470, 545, 626, 709, 810, 933, 1062, 1198, 1362, 1555, 1760, 1980, 2238, 2536, 2858, 3205, 3602, 4063, 4560, 5092, 5704, 6400, 7150, 7966
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Combinatorial interpretation of sequence: [ X1,X2 ] = 2 strictly increasing sequences (possibly null) of odd positive integers; a(n)=#pairs with sum of entries = n.
McKay-Thompson series of class 48g for the Monster group.
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REFERENCES
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T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^2.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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LINKS
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D. Foata and G.-N. Han, Jacobi and Watson Identities Combinatorially Revisited
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FORMULA
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G.f.: 1/(Prod_{k>0} 1+(-x)^k)^2 = (Prod_{k>0} 1+x^(2k-1))^2.
Expansion of q^(1/12)(eta(q^2)^2/(eta(q)eta(q^4)))^2 in powers of q.
Euler transform of period 4 sequence [2, -2, 2, 0, ...].
Expansion of chi(q)^2 in powers of q where chi() is a Ramanujan theta function.
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EXAMPLE
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a(4)=4:[ (1),(3) ],[ (3),(1) ],[ (),(1,3) ],[ (1,3),() ]
T48g = 1/q + 2q^11 + q^23 + 2q^35 + 4q^47 + 4q^59 + 5q^71 + 6q^83 +...
T48g = 1/q + 2*q^11 + q^23 + 2*q^35 + 4*q^47 + 4*q^59 + 5*q^71 + 6*q^83 +...
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(prod(i=1, (1+n)\2, 1+x^(2*i-1), 1+x*O(x^n))^2, n))
(PARI) a(n)=if(n<0, 0, polcoeff(1/prod(i=1, n, 1+(-x)^i, 1+x*O(x^n))^2, n))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)^2/ eta(x+A)/ eta(x^4+A))^2, n))}
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CROSSREFS
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A022597(n)=(-1)^n*a(n).
Sequence in context: A023673 A132965 A022597 this_sequence A134005 A132320 A076369
Adjacent sequences: A073249 A073250 A073251 this_sequence A073253 A073254 A073255
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KEYWORD
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nonn,easy
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AUTHOR
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Michael Somos, Jul 22, 2002
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EXTENSIONS
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Comments from Len Smiley (smiley(AT)math.uaa.alaska.edu).
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