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Search: id:A122135
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| A122135 |
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Expansion of Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k+1))). |
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+0
2
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| 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 26, 31, 40, 48, 60, 72, 89, 106, 130, 154, 186, 220, 264, 310, 370, 433, 512, 598, 704, 818, 958, 1110, 1293, 1494, 1734, 1996, 2308, 2650, 3052, 3496, 4014, 4584, 5248, 5980, 6825, 7760, 8834, 10020, 11380, 12882
(list; graph; listen)
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OFFSET
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0,3
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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(d), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.5). MR0858826 (88b:11063)
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FORMULA
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Euler transform of period 20 sequence [ 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, ...].
G.f.: Sum_{k>=0} x^(k^2+k)/((1-x)(1-x^2)...(1-x^(2k+1))).
Expansion of f(-q^3,-q^7) * f(-q^4,-q^16) / ( f(-q) * f(-q^20) ) in powers of q where f(-q) := f(-q,-q^2) and f(a,b) is Ramanujan's two variable theta function.
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)/ prod(i=1, 2*k+1, 1-x^i, 1+x*O(x^(n-k^2-k)))), n))}
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CROSSREFS
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Adjacent sequences: A122132 A122133 A122134 this_sequence A122136 A122137 A122138
Sequence in context: A082538 A035939 A116665 this_sequence A027194 A039883 A024186
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Aug 21 2006
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