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A007595 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)
15
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; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Number of necklaces of 2 colors with 2n beads and n-1 black ones. - Wouter Meeussen, 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, Nov 13 2004

Number of planar trees (A002995) on n edges with one distinguished edge. - David Callan, Oct 08 2005

Assuming offset 0 this is an analog of A275165: pairs of two Catalan nestings with index sum n. - R. J. Mathar, Jul 19 2016

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..200

Peter J. Cameron, Some treelike objects Quart. J. Math. Oxford Ser. 38 (1987), 155-183. Note that line 3 on p. 163 has a typo. - N. J. A. Sloane, Apr 18 2014

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

A. Gainer-Dewar, PĆ³lya theory for species with an equivariant group action, arXiv preprint arXiv:1401.6202 [math.CO], 2014.

T. Mansour, Counting occurrences of 132 in an even permutation, arXiv:math/0211205 [math.CO], 2002.

FORMULA

G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic, Sep 26 2003

(n+2)*(n+1)*a(n) -6*n*(n+1)*a(n-1) +4*(2*n^2-6*n+1)*a(n-2) +8*(n^2+3*n-7)*a(n-3) -48*(n-2)*(n-3)*a(n-4) +32*(2*n-7)*(n-4)*a(n-5) = 0. - R. J. Mathar, Jun 03 2014

a(n) ~ 4^n /(2*sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Jul 19 2016

a(2n) = A000150(2n). - R. J. Mathar, Jul 19 2016

a(n) = (A000108(n) + 2^n * binomial(1/2, (n+1)/2) * sin(Pi*n/2))/2. - Vladimir Reshetnikov, Oct 03 2016

MAPLE

A007595 := n -> (1/2)*(Cat(n) + (`mod`(n, 2)*Cat((n-1)/2))); Cat := n -> binomial(2*n, n)/(n+1);

MATHEMATICA

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-1)/# ] &)/@Intersection[Divisors[2n], Divisors[n-1]])/(2n), {n, 2, 32}] (* or *) Table[If[EvenQ[n], CatalanNumber[n]/2, (CatalanNumber[n] + CatalanNumber[(n-1)/2])/2], {n, 24}]

Table[(CatalanNumber[n] + 2^n Binomial[1/2, (n + 1)/2] Sin[Pi n/2])/2, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

PROG

(PARI) catalan(n) = binomial(2*n, n)/(n+1);

a(n) = if (n % 2, (catalan(n) + catalan((n-1)/2))/2, catalan(n)/2); \\ Michel Marcus, Jan 23 2016

CROSSREFS

a(n) = A047996(2*n, n-1) for n >= 1 and a(n) = A072506(n, n-1) for n >= 2.

Occurs in A073201 as rows 0, 2, 4, etc. (with a(0)=1 included).

Cf. also A003444, A007123.

Cf. A000150.

Sequence in context: A092566 A036719 A166135 * A148681 A148682 A148683

Adjacent sequences:  A007592 A007593 A007594 * A007596 A007597 A007598

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002

STATUS

approved

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Last modified May 26 05:25 EDT 2017. Contains 287077 sequences.