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Search: id:A122130
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| A122130 |
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Expansion of Sum_{k>0} x^(k^2-1)/((1-x)(1-x^2)...(1-x^(2k-1))). |
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+0
2
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| 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 18, 22, 27, 34, 41, 50, 61, 73, 88, 106, 126, 150, 179, 211, 249, 294, 345, 404, 473, 551, 642, 747, 865, 1002, 1159, 1336, 1539, 1771, 2033, 2331, 2670, 3052, 3485, 3976, 4527, 5150, 5854, 6642, 7530, 8529, 9647, 10902
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
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REFERENCES
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G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(b), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.8). MR0858826 (88b:11063)
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FORMULA
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Expansion of f(-q^2)f(-q^20)/(f(-q)f(-q^8,-q^12)) in powers of q where f(-q)=f(-q,-q^2) and f(a,b) is Ramanujan's two variable theta function.
Euler transform of period 20 sequence [ 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, ...].
G.f.: Sum_{k>0} x^(k^2-1)/((1-x)(1-x^2)...(1-x^(2k-1))).
G,f.: 1/(Product_{k>0} (1-x^(2k-1))(1-x^(20k-8))(1-x^(20k-12))).
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PROGRAM
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(PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=1, sqrtint(n+1), x^(k^2-1)/ prod(i=1, 2*k-1, 1-x^i, 1+x*O(x^(n-k^2+1)))), n))}
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CROSSREFS
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Sequence in context: A134727 A131419 A052816 this_sequence A003073 A123946 A002569
Adjacent sequences: A122127 A122128 A122129 this_sequence A122131 A122132 A122133
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Aug 21 2006, corrected Aug 21 2006
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