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A026837 Number of partitions of n into distinct parts, the greatest being odd.

%I A026837 #20 Feb 17 2021 08:53:25
%S A026837 1,0,1,1,2,2,2,3,4,5,6,8,9,11,13,16,19,23,27,32,38,45,52,61,71,82,96,
%T A026837 111,128,148,170,195,224,256,293,334,380,432,491,556,630,713,805,908,
%U A026837 1024,1152,1295,1455,1632,1829,2049,2291,2560,2859
%N A026837 Number of partitions of n into distinct parts, the greatest being odd.
%C A026837 Fine's theorem: A026838(n) - a(n) = 1 if n = k(3k+1)/2, = -1 if n = k(3k-1)/2, = 0 otherwise (see A143062).
%C A026837 Also number of partitions of n into an odd number of parts and such that parts of every size from 1 to the largest occur. Example: a(9)=4 because we have [3,2,2,1,1],[2,2,2,2,1],[2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 04 2006
%H A026837 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 A026837 G.f.: sum(k>=1, x^(2k-1) * prod(j=1..2k-2, 1+x^j ) ). - _Emeric Deutsch_, Apr 04 2006
%F A026837 a(2*n) = A118302(2*n), a(2*n-1) = A118301(2*n-1); a(n) = A000009(n) - A026838(n). - _Reinhard Zumkeller_, Apr 22 2006
%e A026837 a(9)=4 because we have [9],[7,2],[5,4] and [5,3,1].
%p A026837 g:=sum(x^(2*k-1)*product(1+x^j,j=1..2*k-2),k=1..40): gser:=series(g,x=0,60): seq(coeff(g,x,n),n=1..54); # _Emeric Deutsch_, Apr 04 2006
%t A026837 Table[Count[IntegerPartitions[n],_?(Length[#]==Length[Union[#]] && OddQ[ First[#]]&)],{n,60}] (* _Harvey P. Dale_, Jun 28 2014 *)
%Y A026837 Cf. A026838.
%Y A026837 Cf. A027193.
%K A026837 nonn
%O A026837 1,5
%A A026837 _Clark Kimberling_