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Search: id:A067661
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| A067661 |
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Number of partitions of n into distinct parts such that number of parts is an even number. |
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+0
5
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| 1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404
(list; graph; listen)
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OFFSET
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0,6
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 18 Entry 9 Corollary (2).
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FORMULA
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G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^3 + q^4 + 2 q^5 + 2 q^6 + 3 q^7 + ... = Sum_{n >= 0} q^(n(2n+1))/(q; q)_{2n} (R. William Gosper (rwg(AT)osots.com), Jun 25 2005)
Also, let B(q) = Sum_{n >= 0} A067659(n) q^n = q + q^2 + q^3 + q^4 + q^5 + 2 q^6 + ... Then B(q) = Sum_{n >= 0} q^((n+1)(2n+1))/(q; q)_{2n+1}.
Also we have the following identity involving 2 X 2 matrices:
Prod_{k >= 1} [ 1 q^k / q^k 1 ] = [ A(q) B(q) / B(q) A(q) ] (R. William Gosper (rwg(AT)osots.com), Jun 25 2005)
a(n) = (A000009(n)+A010815(n))/2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 24 2002
Expansion of (1+phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos Feb 14 2006
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PROGRAM
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(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A)+eta(x+A))/2, n))} /* Michael Somos Feb 14 2006 */
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CROSSREFS
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Sequence in context: A036821 A026798 A125890 this_sequence A052839 A125894 A091493
Adjacent sequences: A067658 A067659 A067660 this_sequence A067662 A067663 A067664
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KEYWORD
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easy,nonn
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AUTHOR
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Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 23 2002
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