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Search: id:A026838
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| A026838 |
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Number of partitions of n into distinct parts, the greatest being even. |
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+0
6
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| 0, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Fine's theorem: a(n) - A026837(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 even number of parts and such that parts of every size from 1 to the largest occur. Example: a(8)=3 because we have [3,2,2,1],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2006
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REFERENCES
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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
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G.f.=sum(x^(2k)*product(1+x^j, j=1..2k-1), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2006
a(2*n)=A118301(2*n), a(2*n-1)=A118302(2*n-1); a(n)=A000009(n)-A026837(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 22 2006
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EXAMPLE
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a(8)=3 because we have [8],[6,2], and [4,3,1].
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MAPLE
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g:=sum(x^(2*k)*product(1+x^j, j=1..2*k-1), k=1..50): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=1..54); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2006
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CROSSREFS
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Cf. A026837.
Cf. A027187.
Sequence in context: A039732 A011876 A029036 this_sequence A017864 A029035 A118302
Adjacent sequences: A026835 A026836 A026837 this_sequence A026839 A026840 A026841
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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