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Number of partitions of n with even number (or 0) 2's.
5

%I #16 Oct 30 2015 11:24:49

%S 1,1,1,2,4,5,7,10,15,20,27,36,50,65,85,111,146,186,239,304,388,488,

%T 614,767,961,1191,1475,1819,2243,2746,3361,4096,4988,6047,7322,8836,

%U 10655,12801,15360,18384,21978,26199,31196,37062,43979,52072,61579,72682

%N Number of partitions of n with even number (or 0) 2's.

%H Vaclav Kotesovec, <a href="/A092295/b092295.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = A000041(n)-a(n-2).

%F G.f.=1/[(1+x^2)*product(1-x^j, j=1..infinity)]. - _Emeric Deutsch_, Mar 30 2006

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Oct 30 2015

%e a(5)=5 because the partitions [5],[4,1],[3,1,1],[2,2,1] and [1,1,1,1,1] of 5 have an even number of 2's ([3,2] and [2,1,1,1] do not qualify).

%p g:=1/(1+x^2)/product(1-x^j,j=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..47); # _Emeric Deutsch_, Mar 30 2006

%t nmax = 50; CoefficientList[Series[1/((1+x^2) * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 30 2015 *)

%Y Cf. A087787.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Feb 06 2004

%E More terms from _Benoit Cloitre_, Feb 08 2004