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Search: id:A035294
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| A035294 |
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Number of ways to partition 2n into distinct positive integers. |
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+0
11
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic (vladeta(AT)Eunet.yu), 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 (reinhard.zumkeller(AT)lhsystems.com). Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.
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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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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 (vladeta(AT)Eunet.yu), 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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PROGRAM
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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: A032417 A073470 A086182 this_sequence A073818 A116084 A108925
Adjacent sequences: A035291 A035292 A035293 this_sequence A035295 A035296 A035297
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KEYWORD
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nonn
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AUTHOR
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njas, R. W. Gosper
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