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A035294
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Number of ways to partition 2n into distinct positive integers.
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12
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1, 1, 2, 4, 6, 10, 15, 22, 32, 46, 64, 89, 122, 165, 222, 296, 390, 512, 668, 864, 1113, 1426, 1816, 2304, 2910, 3658, 4582, 5718, 7108, 8808, 10880, 13394, 16444, 20132, 24576, 29927, 36352, 44046, 53250, 64234, 77312, 92864, 111322, 133184, 159046
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OFFSET
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0,3
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COMMENTS
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Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic, Aug 17 2004
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. Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.
Also, number of partitions of 2n with max descent 1 and last part 1. - Wouter Meeussen, Mar 31 2013
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.
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LINKS
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Table of n, a(n) for n=0..44.
N. J. A. Sloane, Transforms
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FORMULA
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Expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ).
a(n) = t(2*n, 0), t as defined in A079211.
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, Oct 10 2004
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(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).
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
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EXAMPLE
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a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].
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MATHEMATICA
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Table[Count[Partitions[2 n], q_ /; Union[q] == Sort[q]], {n, 16}];
Table[Count[Partitions[2 n], q_ /; Count[q, _?EvenQ] == 0], {n, 16}];
Table[Count[Partitions[2 n], q_ /; Last[q] == 1 && Max[q - PadRight[Rest[q], Length[q]]] <= 1 ], {n, 16}];
(* Wouter Meeussen, Mar 31 2013 *)
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PROG
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(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))}
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CROSSREFS
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Cf. A078408, A078406, A078407. a(n)=A000009(2n).
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.
Sequence in context: A175826 A073470 A086182 * A073818 A143184 A116084
Adjacent sequences: A035291 A035292 A035293 * A035295 A035296 A035297
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, R. W. Gosper
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STATUS
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approved
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