0,5

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)lhsystems.com), Apr 22 2006

M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

T. D. Noe, Table of n, a(n) for n=0..1000

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.yu), 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

with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;

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

nonn,easy,nice

njas

Better description and more terms from Christian G. Bower (bowerc(AT)usa.net), Apr 27, 2000.