A000150 Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

0, 0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220, 4847208, 17678835, 64821680, 238819350, 883629164, 3282060210, 12233125112, 45741281820, 171529777432, 644952073662, 2430973096720, 9183676536076

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Number of Dyck paths of length 2n having an odd number of peaks at even height. Example: a(3)=2 because we have UDU(UD)D and U(UD)DUD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004

For n>=1, a(n) is the number of unordered binary trees with n internal nodes in which the left subtree is distinct from the right subtree. - Geoffrey Critzer, Feb 21 2013

Assuming offset -1 this is an analog of A275166: pairs of distinct Catalan numbers with index sum n. - R. J. Mathar, Jul 19 2016

References

• S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
• R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
• R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 78, (3.5.26).
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

Links

• T. D. Noe, Table of n, a(n) for n=0..200
• S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
• R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
• F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
• Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.
• P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]
• Index entries for sequences related to Lyndon words

Crossrefs

a(n) = T(2n+2, n), array T as in A051168, a count of Lyndon words.

Cf. A051168, A005430.

Cf. A007595.

A diagonal of the square array described in A051168.

Programs

Mathematica

nn=20;CoefficientList[Series[x/2(((1-(1-4x)^(1/2))/(2x))^2-(1-(1-4x^2)^(1/2))/(2x^2)),{x,0,nn}],x]  (* Geoffrey Critzer, Feb 21 2013 *)

Formula

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (c(x)-d(x))/2.

G.f.: (sqrt(1-4*z^2) - sqrt(1-4*z) - 2*z)/(4*z). - Emeric Deutsch, Nov 13 2004

With c(x) defined as above: g.f. = x*(c(x)^2/2 - c(x^2)/2). - Geoffrey Critzer, Feb 21 2013

a(n) = ( 2^(n-3)/sqrt(Pi) ) * ( 4*2^n*GAMMA(n+1/2)/GAMMA(n+2) + ((-1)^n - 1)*GAMMA(n/2)/GAMMA(n/2 + 3/2) ) for n>0. - Mark van Hoeij, Nov 11 2009

a(n) ~ 2^(2*n-1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2014

a(2n) = A000108(2n) / 2; a(2n+1) = ( A000108(2n+1) - A000108(n) ) / 2. - John Bodeen, Jun 24 2015

D-finite with recurrence +n*(n+1)*(n-2)^2*a(n) -2*n*(2*n-5)*(n-1)^2*a(n-1) -4*n*(n-2)^3*a(n-2) +8*(2*n-5)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Oct 28 2021

Extensions

Additional comments from Clark Kimberling