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Number of partitions of n into distinct parts, the greatest being even.
10

%I #24 Feb 17 2021 08:53:42

%S 0,1,1,1,1,2,3,3,4,5,6,7,9,11,14,16,19,23,27,32,38,44,52,61,71,83,96,

%T 111,128,148,170,195,224,256,292,334,380,432,491,557,630,713,805,908,

%U 1024,1152,1295,1455,1632,1829,2048,2291,2560,2859

%N Number of partitions of n into distinct parts, the greatest being even.

%C Fine's theorem: a(n) - A026837(n) = 1 if n = k(3k+1)/2, = -1 if n = k(3k-1)/2, = 0 otherwise (see A143062).

%C Also number of partitions of n into an even number of parts and such that parts of every size from 1 to the largest occur. Example: a(8)=3 because we have [3,2,2,1], [2,2,1,1,1,1] and [1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 04 2006

%H Vincenzo Librandi, <a href="/A026838/b026838.txt">Table of n, a(n) for n = 1..1000</a>

%H I. Pak, <a href="http://www.math.ucla.edu/~pak/papers/finefull.pdf">On Fine's partition theorems, Dyson, Andrews and missed opportunities</a>, Math. Intelligencer, 25 (No. 1, 2003), 10-16.

%F G.f.: sum(k>=1, x^(2k) * prod(j=1..2k-1, 1+x^j ) ). - _Emeric Deutsch_, Apr 04 2006

%F a(2*n) = A118301(2*n), a(2*n-1) = A118302(2*n-1); a(n) = A000009(n) - A026837(n). - _Reinhard Zumkeller_, Apr 22 2006

%e a(8)=3 because we have [8],[6,2] and [4,3,1].

%p g:=sum(x^(2*k)*product(1+x^j,j=1..2*k-1),k=1..50): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..54); # _Emeric Deutsch_, Apr 04 2006

%t nn=54;CoefficientList[Series[Sum[x^(2j)Product[1+ x^i,{i,1,2j-1}],{j,0,nn}],{x,0,nn}],x] (* _Geoffrey Critzer_, Jun 20 2014 *)

%Y Cf. A026837, A027187.

%K nonn

%O 1,6

%A _Clark Kimberling_