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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) 16
 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, 9183676536076, 34766775829452, 131873975875180 (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., Vol. 38, No. 2 (1987), pp. 155-183. Note that line 3 on p. 163 has a typo. - N. J. A. Sloane, Apr 18 2014 Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seq., Vol. 3 (2000), Article 00.1.5. Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.4. Andrew Gainer-Dewar, Pólya theory for species with an equivariant group action, arXiv preprint arXiv:1401.6202 [math.CO], 2014. Toufik Mansour, Counting occurrences of 132 in an even permutation, arXiv:math/0211205 [math.CO], 2002. Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022. FORMULA G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic, Sep 26 2003 D-finite with recurrence: n*(n+1)*a(n) -6*n*(n-1)*a(n-1) +4*(2*n^2-10*n+9)*a(n-2) +8*(n^2+n-9)*a(n-3) -48*(n-3)*(n-4)*a(n-4) +32*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Jun 03 2014, adapted to offset Feb 20 2020 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 Sum_{n>=1} a(n)/4^n = (3-sqrt(3))/2 (A334843). - Amiram Eldar, Mar 20 2022 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 *) Table[If[EvenQ[n], CatalanNumber[n]/2, (CatalanNumber[n]+CatalanNumber[(n-1)/2])/2], {n, 30}] (* Harvey P. Dale, Sep 06 2021 *) 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. A000108, A000150, A334843. 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 December 6 17:01 EST 2023. Contains 367612 sequences. (Running on oeis4.)