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Search: id:A092306
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| A092306 |
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Number of partitions of n such that the set of parts has an even number of elements. |
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+0
3
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| 0, 0, 1, 2, 5, 6, 11, 13, 17, 23, 29, 34, 47, 64, 74, 107, 136, 185, 233, 308, 392, 518, 637, 814, 1002, 1272, 1560, 1912, 2339, 2863, 3475, 4212, 5123, 6147, 7398, 8935, 10734, 12843, 15464, 18382, 22041, 26249, 31326, 37213, 44273, 52375, 62103, 73376
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OFFSET
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1,4
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FORMULA
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a(n) = b(n, 1, 0, 1) with b(n, i, j, f) = if i<n then b(n-i, i, i, 1-f-(1-2*f)*0^(i-j)) + b(n, i+1, j, f) else (1-f-(1-2*f)*0^(i-j))*0^(i-n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1+Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)+A104575(n))/2.
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EXAMPLE
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The partitions of five are: {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}, The seven partitions have 1, 2, 2, 2, 2, 2 and 1 distinct parts respectfully.
n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2}, and {1}, six of them have an even number of elements, therefore a(6)=6.
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MATHEMATICA
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first Needs["DiscreteMath`Combinatorica`"], then f[n_] := Count[ Mod[ Length /@ Union /@ Partitions[n], 2], 0]; Table[ f[n], {n, 1, 49}] (from Robert G. Wilson v Feb 16 2004)
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CROSSREFS
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Cf. A060177, A002133, A027187, A090794.
Adjacent sequences: A092303 A092304 A092305 this_sequence A092307 A092308 A092309
Sequence in context: A030130 A045845 A002133 this_sequence A090552 A024520 A015613
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KEYWORD
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nonn
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AUTHOR
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Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 12 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 16 2004
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