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A133153
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Number of partitions of n into parts that are odd or == +- 2 mod 10.
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0
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1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 71, 90, 112, 140, 174, 214, 263, 322, 392, 476, 576, 694, 834, 1000, 1194, 1423, 1692, 2005, 2372, 2800, 3296, 3874, 4544, 5318, 6214, 7248, 8438, 9808, 11383, 13188, 15258, 17628, 20334, 23426, 26952, 30966, 35536, 40730
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Andrews gives a second interpration of these numbers and refers to them as a cousin of the Rogers-Ramanujan numbers.
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, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 10.)
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 7, Eq. (1.3). MR0858826 (88b:11063)
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FORMULA
| Expansion of f(-q^4, -q^6) / f(-q) in powers of q where f() is Ramanujan's theta function.
Euler transform of period 10 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...]. - Michael Somos Sep 30 2007
G.f.: (Sum_{k>=0} x^(2*k^2) / ( (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+1)/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
| 1 + q + 2*q^2 + 3*q^3 + 4*q^4 + 6*q^5 + 8*q^6 + 11*q^7 + 15*q^8 + ...
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MATHEMATICA
| th[m_][a_, b_] := Sum[a^(n*((n+1)/2))*b^(n*((n-1)/2)), {n, -m, m}]; th[m_][-q_] := th[m][-q, -q^2];
f[s_List] := (m++; CoefficientList[ Series[ th[m][-q^4, -q^6]/th[m][-q], {q, 0, 51}], q]); FixedPoint[f, m = 0; {}] (* From Jean-François Alcover, Jul 19 2011 *)
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PROG
| (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) / (1 - x^k) / (1 - x^(2*k + 1)) + x * O(x^n), t), n))} /* Michael Somos Sep 30 2007 */
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CROSSREFS
| Sequence in context: A175868 A035950 A175864 * A100673 A115671 A105782
Adjacent sequences: A133150 A133151 A133152 * A133154 A133155 A133156
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 22 2007
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