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A122129
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Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).
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1
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1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 30, 37, 46, 57, 69, 84, 102, 123, 148, 177, 211, 252, 299, 353, 417, 491, 576, 675, 789, 920, 1071, 1244, 1442, 1670, 1929, 2224, 2562, 2946, 3381, 3876, 4437, 5072, 5791, 6602, 7517, 8551, 9714, 11021, 12493
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Generating function arises naturally in Rodney Baxter's solution of the Hard Hexagon Model according to George Andrews.
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REFERENCES
| G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1999; Exercise 6(a), p. 591.
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 8, Eq. (1.7). MR0858826 (88b:11063)
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FORMULA
| Euler transform of period 20 sequence [ 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, ...].
Expansion of f(-q^2)f(-q^20)/(f(-q)f(-q^4,-q^16)) in powers of q where f(-q)=f(-q,-q^2) and f(a,b) is Ramanujan's two variable theta function.
G.f.: 1 +Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).
G.f. 1/(Product_{k>0} (1-x^(2k-1))(1-x^(20k-4))(1-x^(20k-16))).
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PROG
| (PARI) {a(n)=if(n<1, n==0, polcoeff( sum(k=1, sqrtint(n), x^k^2/prod(i=1, 2*k, 1-x^i, 1+x*O(x^(n-k^2)))), n))}
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CROSSREFS
| Sequence in context: A190269 A039853 A062188 * A003413 A100853 A174065
Adjacent sequences: A122126 A122127 A122128 * A122130 A122131 A122132
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Aug 21 2006
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