Search: id:A026837
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%I A026837
%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,
%T A026837 82,96,111,128,148,170,195,224,256,293,334,380,432,491,556,630,
%U A026837 713,805,908,1024,1152,1295,1455,1632,1829,2049,2291,2560,2859
%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.
%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 (deutsch(AT)duke.poly.edu), Apr 
               04 2006
%D A026837 I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, 
               Math. Intelligencer, 25 (No. 1, 2003), 10-16.
%F A026837 G.f.=sum(x^(2k-1)*product(1+x^j, j=1..2k-2), k=1..infinity). - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), 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 (reinhard.zumkeller(AT)gmail.com), 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 (deutsch(AT)duke.poly.edu), 
               Apr 04 2006
%Y A026837 Cf. A026838.
%Y A026837 Cf. A027193.
%Y A026837 Sequence in context: A018121 A111212 A102240 this_sequence A005855 A096748 
               A022866
%Y A026837 Adjacent sequences: A026834 A026835 A026836 this_sequence A026838 A026839 
               A026840
%K A026837 nonn
%O A026837 1,5
%A A026837 Clark Kimberling (ck6(AT)evansville.edu)

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