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Search: id:A007595
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| A007595 |
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a(n) = C_n / 2 if n is even or ( C_n + C_((n-1)/2) ) / 2 if n is odd, where C = Catalan numbers (A000108).
(Formerly M2681)
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+0
11
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| 1, 1, 3, 7, 22, 66, 217, 715, 2438, 8398, 29414, 104006, 371516, 1337220, 4847637, 17678835, 64823110, 238819350, 883634026, 3282060210, 12233141908, 45741281820, 171529836218, 644952073662, 2430973304732
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Number of necklaces of 2 colors with 2n beads and n-1 black ones. - Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 03 2002
Number of rooted planar binary trees up to reflection (trees with n internal nodes, or a total of 2n+1 nodes). - Antti Karttunen, Aug 19 2002
Number of even permutations avoiding 132.
Number of Dyck paths of length 2n having an even number of peaks at even height. Example: a(3)=3 because we have UDUDUD, U(UD)(UD)D and UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2004
Number of planar trees (A002995) on n edges with one distinguished edge. - David Callan (callan(AT)stat.wisc.edu), Oct 08 2005
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..200
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
T. Mansour, Counting occurrences of 132 in an even permutation.
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FORMULA
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G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 26 2003
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MAPLE
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A007595 := n -> (1/2)*(Cat(n) + (`mod`(n, 2)*Cat((n-1)/2))); Cat := n -> binomial(2*n, n)/(n+1);
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MATHEMATICA
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Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-1)/# ] &)/@Intersection[Divisors[2n], Divisors[n-1]])/(2n), {n, 2, 32}] or Table[If[EvenQ[n], cat[n]/2, (cat[n] +cat[(n-1)/2])/2], {n, 24}] with cat[n]=A000108
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CROSSREFS
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a(n)=A047996(2*n, n-1) for n>= 1 and a(n)=A072506(n, n-1) for n>=2. Occurs in A073201 as the rows 0, 2, 4, etc. (with a(0)=1 included). Cf. also A003444, A007123.
Cf. A000150.
Sequence in context: A092566 A036719 A166135 this_sequence A148681 A148682 A148683
Adjacent sequences: A007592 A007593 A007594 this_sequence A007596 A007597 A007598
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002.
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