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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).

%I A143024 #9 Jul 23 2019 02:05:51
%S A143024 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,
%T A143024 720,224,28,0,0,0,0,0,0,3876,7752,6783,3192,798,84,0,0,0,0,0,0,0,
%U A143024 21318,52668,58520,36960,13860,2904,264,0,0,0,0,0,0,0,0,120175,354200,478170
%N 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).
%C A143024 Row n contains 2n-4 terms, the first n-2 of which are 0.
%C A143024 Row sums yield A089436.
%C A143024 T(n,n-1) = A006013(n-2).
%C A143024 Sum_{k=2..2n-4} k*T(n,k) = A143025.
%H A143024 C. Domb and A. J. Barrett, <a href="http://dx.doi.org/10.1016/0012-365X(74)90081-8">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358.
%H A143024 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%F A143024 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).
%F A143024 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)).
%e A143024 T(3,2)=2 because we have {AB,BC} and {AC, BC} (A is the root).
%e A143024 Triangle starts:
%e A143024   1;
%e A143024   0,   2;
%e A143024   0,   0,   7,   2;
%e A143024   0,   0,   0,  30,  20,   4;
%e A143024   0,   0,   0,   0, 143, 156,  65,  10;
%p A143024 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
%Y A143024 Cf. A007297, A089436, A006013, A143025.
%K A143024 nonn,tabf
%O A143024 2,3
%A A143024 _Emeric Deutsch_, Jul 31 2008