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Search: id:A000701
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| A000701 |
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One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.
(Formerly M0645 N0239)
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+0
17
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| 0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Also number of cycle types of odd permutations.
Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - Naoki Sato (nsato7(AT)yahoo.ca), Jul 20 2005. E.g. a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2006
Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
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REFERENCES
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M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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a(n)=(A000041(n)-A000700(n))/2.
Generating functions from R. William Gosper (rwg(AT)osots.com), Aug 08 2005:
Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...
= -( sum_{n = 1 .. oo} (-q^2)^(n^2) ) / ( sum_{ n = -oo, oo } (-1)^n q^(n(3n-1)/2) )
= (- q; q)_{oo} sum_{n=1..oo} q^(2(2n-1))/(q^2;q^2)_{2n-1}
= (1/(q;q)_oo - 1/(q;-q)_oo)/2
= (1/(q;q)_oo - (-q;q^2)_oo)/2
= sum{ k = 0..oo } ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2
using the "q-pochhammer" notation (a;q)_n := prod_{k=0..n-1} 1-a*q^k.
a(n) = p(n-2)-p(n-8)+p(n-18)-p(n-32)+... +(-1)^(k+1)*p(n-2*k^2)+..., where p() is A000041(). E.g. a(20) = p(18)-p(12)+p(2) = 385-77+2 = 310. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 08 2004
G.f.=(1/2)(1-product((1-x^(2j))/(1+x^(2j)), j=1..infinity))/product(1-x^j, j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2006
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MAPLE
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with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;
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CROSSREFS
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Cf. A000041, A000700, A046682.
Cf. A118302.
Sequence in context: A036005 A104503 A027340 this_sequence A123975 A094984 A107332
Adjacent sequences: A000698 A000699 A000700 this_sequence A000702 A000703 A000704
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Better description and more terms from Christian G. Bower (bowerc(AT)usa.net), Apr 27, 2000.
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