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Search: id:A067357
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| A067357 |
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Number of self-conjugate partitions of 4n+1 into odd parts. |
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1
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| 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 8, 10, 10, 12, 14, 15, 18, 20, 22, 26, 29, 32, 36, 40, 44, 50, 56, 60, 68, 76, 82, 92, 101, 110, 122, 134, 146, 160, 176, 191, 210, 230, 248, 272, 296, 320, 350, 380, 410, 446, 484, 522, 566, 612, 660, 715, 772, 830, 896, 966, 1038, 1120
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Also number of partitions of n in which even parts are distinct and if k occurs then so does every positive even number less than k (Dean Hickerson). Absolute values of the terms of A053254. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 10 2006
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REFERENCES
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P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 260, Article 512.
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FORMULA
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G.f.=sum(q^(k(k+1))/product(1-q^(2j+1),j=0..k),k=0..infinity) (Emeric Deutsch and Dean Hickerson). G.f.=sum(q^n*product(1+q^(2*j-1),j=1..n),n=0..infinity) (Dean Hickerson and Vladeta Jovovic).
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EXAMPLE
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a(5)=3 because we have [11,1,1,1,1,1,1,1,1,1,1], [9,3,3,1,1,1,1,1,1] and [5,5,5,3,3].
1 + q + 2*q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
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MAPLE
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g:=sum(q^(k*(k+1))/product(1-q^(2*j+1), j=0..k), k=0..8): gser:=series(g, q=0, 80): seq(coeff(gser, q, n), n=0..75); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 10 2006
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PROGRAM
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{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 4*n+1) -1) \ 2, x^(k^2 + k) / prod(j=0, k, 1 - x^(2*j+1), 1 + x * O(x^(n - k^2 - k)))), n))} /* Michael Somos Jan 27 2008 */
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CROSSREFS
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Cf. A000700, A000701.
Cf. A053254.
Sequence in context: A029146 A029053 A053254 this_sequence A051059 A132967 A132968
Adjacent sequences: A067354 A067355 A067356 this_sequence A067358 A067359 A067360
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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 24 2002
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 10 2006
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