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A026837
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Number of partitions of n into distinct parts, the greatest being odd.
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6
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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, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Fine's theorem: A026838(n) - a(n) = 1 if n = k(3k+1)/2, = -1 if n = k(3k-1)/2, = 0 otherwise.
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
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REFERENCES
| I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.
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FORMULA
| 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
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
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EXAMPLE
| a(9)=4 because we have [9],[7,2],[5,4] and [5,3,1].
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MAPLE
| 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
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CROSSREFS
| Cf. A026838.
Cf. A027193.
Sequence in context: A018121 A111212 A102240 * A005855 A096748 A022866
Adjacent sequences: A026834 A026835 A026836 * A026838 A026839 A026840
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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