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Search: id:A134157
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| A134157 |
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Number of partitions of n into parts that are odd or == +- 4 mod 10. |
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+0
2
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| 1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 23, 30, 38, 48, 61, 76, 94, 117, 144, 176, 216, 262, 317, 384, 462, 554, 664, 792, 942, 1120, 1326, 1566, 1848, 2174, 2552, 2992, 3499, 4084, 4762, 5540, 6434, 7464, 8642, 9991, 11538, 13302, 15314, 17612, 20225, 23196
(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, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 7, Eq. (1.4). MR0858826 (88b:11063)
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FORMULA
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Expansion of f(-q^2, -q^8) / f(-q) in powers of q where f() is Ramanujan's theta function.
Euler transform of period 10 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].
G.f.: (Sum_{k>=0} x^(2*(k^2+k)) / ( (1 - x^2) * (1 - x^4) * ... * (1 - x^(2*k)) ) )/ Product_{k>0} 1 - x^(2*k-1).
G.f.: Sum_{k>=0} x^(k*(3*k+3)/2) * (1 + x) * (1 + x^2) * ... * (1 + x^(2*k)) / ( (1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1)) ).
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EXAMPLE
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1 + q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 6*q^6 + 8*q^7 + 10*q^8 + 14*q^9 + ...
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PROGRAM
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(PARI) {a(n) = local(t); if( n<0, 0, t = 1 / (1 - x) + x * O(x^n); polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1)\6, t = t * x^(3*k) / (1 - x^k) / (1 - x^(2*k + 1)) + x * O(x^n), t), n))}
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CROSSREFS
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Adjacent sequences: A134154 A134155 A134156 this_sequence A134158 A134159 A134160
Sequence in context: A053253 A095913 A102848 this_sequence A045476 A066816 A123586
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Oct 10 2007
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