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Search: id:A027193
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| A027193 |
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Number of partitions of n into an even number of parts, the least being 1; also, a(n+1) = number of partitions of n into an odd number of parts; also, partitions of n+1 in which greatest part is odd. |
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+0
12
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| 0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Also number of partitions of n such that the largest part is even and occurs only once. Example: a(6)=4 because we have [6],[4,2],[4,1,1] and [2,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2006
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
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FORMULA
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a(n+1)=(A000041(n)-(-1)^n*A000700(n))/2.
For g.f. see under A027187.
G.f.=sum(x^(2k)/product(1-x^j,j=1..2k-1),k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2006
a(2*n)=A000701(2*n), a(2*n-1)=A046682(2*n-1); a(n)=A000041(n)-A027187(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006
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EXAMPLE
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a(6)=4 because we have [5,1],[3,1,1,1],[2,2,1,1] and [1,1,1,1,1,1].
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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..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=1..45); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2006
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CROSSREFS
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Cf. A027187, A000701, A046682.
Cf. A026837.
Sequence in context: A036002 A104504 A027337 this_sequence A126796 A157162 A109434
Adjacent sequences: A027190 A027191 A027192 this_sequence A027194 A027195 A027196
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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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