Search: id:A035294
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%I A035294
%S A035294 1,1,2,4,6,10,15,22,32,46,64,89,122,165,222,296,390,512,668,864,1113,
%T A035294 1426,1816,2304,2910,3658,4582,5718,7108,8808,10880,13394,16444,20132,
%U A035294 24576,29927,36352,44046,53250,64234,77312,92864,111322,133184,159046
%N A035294 Number of ways to partition 2n into distinct positive integers.
%C A035294 Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Aug 17 2004
%C A035294 This sequence was originally defined as the expansion of sum ( q^n / 
               product( 1-q^k, k=1..2*n), n=0..inf ). The present definition is 
               due to Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com). Michael 
               Somos points out that the equivalence of the two definitions follows 
               from Andrews, page 19.
%D A035294 G. E. Andrews, The Theory of Partitions, Cambridge University Press, 
               1998, p. 19.
%H A035294 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A035294 Expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ).
%F A035294 a(n) = t(2*n, 0), t as defined in A079211.
%F A035294 G.f.: Product((1 + x^(8 * i + 1)) * (1 + x^(8 * i + 2))^2 * (1 + x^(8 
               * i + 3))^2 * (1 + x^(8 * i + 4))^3 * (1 + x^(8 * i + 5))^2 * (1 
               + x^(8 * i + 6))^2 * (1 + x^(8 * i + 7)) * (1 + x^(8 * i + 8))^3, 
               i=0..infinity). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 10 2004
%F A035294 G.f.: (Sum_{k>=0} x^A074378(k))/(Product_{k>0} (1-x^k)) = f(x^3, x^5)/
               f(-x) . - Michael Somos Nov 01 2005
%F A035294 f(a,b)=Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable 
               theta function and f(-x)=f(-x,-x^2).
%F A035294 Euler transform of period 16 sequence [ 1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 
               2, 1, 2, 1, 1, 0, ...]. - Michael Somos Aug 16 2007
%e A035294 a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].
%o A035294 (PARI) {a(n)=local(A); if(n<0, 0, n*=2; A=x*O(x^n); polcoeff( eta(x^2+A)/
               eta(x+A), n))}
%Y A035294 Cf. A078408, A078406, A078407. a(n)=A000009(2n).
%Y A035294 Cf. A079122, A079126, A079124, A079125, A067953.
%Y A035294 Cf. A005408.
%Y A035294 Sequence in context: A152415 A073470 A086182 this_sequence A073818 A143184 
               A116084
%Y A035294 Adjacent sequences: A035291 A035292 A035293 this_sequence A035295 A035296 
               A035297
%K A035294 nonn
%O A035294 0,3
%A A035294 N. J. A. Sloane (njas(AT)research.att.com), R. W. Gosper

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