A143024 Triangle read by rows: T(n,k) is the number of non-crossing connected graphs on n nodes on a circle having root (a distinguished node) of degree 1 and having k edges (n >= 2, 1 <= k <= 2n-4).

1, 0, 2, 0, 0, 7, 2, 0, 0, 0, 30, 20, 4, 0, 0, 0, 0, 143, 156, 65, 10, 0, 0, 0, 0, 0, 728, 1120, 720, 224, 28, 0, 0, 0, 0, 0, 0, 3876, 7752, 6783, 3192, 798, 84, 0, 0, 0, 0, 0, 0, 0, 21318, 52668, 58520, 36960, 13860, 2904, 264, 0, 0, 0, 0, 0, 0, 0, 0, 120175, 354200, 478170

Offset

2,3

Comments

Row n contains 2n-4 terms, the first n-2 of which are 0.

Row sums yield A089436.

T(n,n-1) = A006013(n-2).

Sum_{k=2..2n-4} k*T(n,k) = A143025.

Links

Table of n, a(n) for n=2..69.

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

Formula

T(n,k) = 2*binomial(k-2, n-3)*binomial(3n-5, 2n-k-4)/(n-2) (n >= 3, 2 <= k <= 2n-4); T(2,1)=1; T(2,k)=0 (k >= 2).

The trivariate g.f. G=G(t,s,z) for non-crossing connected graphs on nodes on a circle, with respect to number of nodes (marked by z), number of edges (marked by t) and degree of root (marked by s) is G=z + tszg^2/[z-ts(g - z + g^2)], where g=g(t,z) satisfies tg^3 + tg^2 - (1 + 2t)zg +(1 + t)z^2 = 0 (see Domb & Barrett, Eq. (47); Flajolet & Noy, Eq. (18)).

Example

T(3,2)=2 because we have {AB,BC} and {AC, BC} (A is the root).
Triangle starts:
  1;
  0,   2;
  0,   0,   7,   2;
  0,   0,   0,  30,  20,   4;
  0,   0,   0,   0, 143, 156,  65,  10;

Maple

T:=proc(n, k) options operator, arrow: 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4)/(n-2) end proc: 1; for n from 3 to 10 do 0, seq(T(n, k), k=2..2*n-4) end do; % yields sequence in triangular form

Crossrefs

Cf. A007297, A089436, A006013, A143025.

Sequence in context: A100344 A370796 A094596 * A271971 A278157 A198232

Adjacent sequences: A143021 A143022 A143023 * A143025 A143026 A143027

Keyword

nonn,tabf

Author

Emeric Deutsch, Jul 31 2008

Status

approved