Search: id:A000701
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%I A000701 M0645 N0239
%S A000701 0,0,1,1,2,3,5,7,10,14,20,27,37,49,66,86,113,146,190,242,310,392,497,623,
%T A000701 782,973,1212,1498,1851,2274,2793,3411,4163,5059,6142,7427,8972,10801,
%U A000701 12989,15572,18646,22267,26561,31602,37556,44533,52743,62338,73593
%N A000701 One half of number of non-self-conjugate partitions; also half of number 
               of asymmetric Ferrers graphs with n nodes.
%C A000701 Also number of cycle types of odd permutations.
%C A000701 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
%C A000701 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
%D A000701 M. Osima, On the irreducible representations of the symmetric group, 
               Canad. J. Math., 4 (1952), 381-384.
%D A000701 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000701 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000701 T. D. Noe, <a href="b000701.txt">Table of n, a(n) for n=0..1000</a>
%F A000701 a(n)=(A000041(n)-A000700(n))/2.
%F A000701 Generating functions from R. William Gosper (rwg(AT)osots.com), Aug 08 
               2005:
%F A000701 Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...
%F A000701 = -( sum_{n = 1 .. oo} (-q^2)^(n^2) ) / ( sum_{ n = -oo, oo } (-1)^n 
               q^(n(3n-1)/2) )
%F A000701 = (- q; q)_{oo} sum_{n=1..oo} q^(2(2n-1))/(q^2;q^2)_{2n-1}
%F A000701 = (1/(q;q)_oo - 1/(q;-q)_oo)/2
%F A000701 = (1/(q;q)_oo - (-q;q^2)_oo)/2
%F A000701 = sum{ k = 0..oo } ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2
%F A000701 using the "q-pochhammer" notation (a;q)_n := prod_{k=0..n-1} 1-a*q^k.
%F A000701 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
%F A000701 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
%p A000701 with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;
%Y A000701 Cf. A000041, A000700, A046682.
%Y A000701 Cf. A118302.
%Y A000701 Sequence in context: A036005 A104503 A027340 this_sequence A123975 A094984 
               A107332
%Y A000701 Adjacent sequences: A000698 A000699 A000700 this_sequence A000702 A000703 
               A000704
%K A000701 nonn,easy,nice
%O A000701 0,5
%A A000701 N. J. A. Sloane (njas(AT)research.att.com).
%E A000701 Better description and more terms from Christian G. Bower (bowerc(AT)usa.net), 
               Apr 27, 2000.

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