A127537 - OEIS

%I #19 Jul 29 2018 08:09:28

%S 1,0,3,1,0,0,12,9,2,0,0,0,55,66,30,5,0,0,0,0,273,455,315,105,14,0,0,0,

%T 0,0,1428,3060,2856,1428,378,42,0,0,0,0,0,0,7752,20349,23940,15960,

%U 6300,1386,132,0,0,0,0,0,0,0,43263,134596,191268,159390,83490,27324,5148,429

%N Triangle read by rows: T(n,k) (n >= 2, 1 <= k <= 2n-3) is the number of non-crossing connected graphs on n nodes on a circle, having k edges. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....

%C Row n contains 2n-3 terms, the first n-2 of which are equal to 0.

%C T(n,n-1) = A001764(n-1). T(n,2n-3) = A000108(n-2) (the Catalan numbers).

%C T(n,k) = A089434(n,k+1-n).

%C Sum_{k=n-1..2n-3} k*T(n,k) = A045741(n).

%C Sum_{n=k..2k-2} T(n,k) = A065065(k).

%H 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 C. Domb & A. J. Barrett, <a href="/A003408/a003408.pdf">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)

%H C. Domb & A. J. Barrett, <a href="/A001764/a001764.pdf">Notes on Table 2 in "Enumeration of ladder graphs"</a>, Discrete Math. 9 (1974), 55. (Annotated scanned copy)

%H 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 T(n,k) = C(3n-3,n+k)C(k-1,k-n+1)/(n-1) (n >= 2, 0 <= k <= 2n-3).

%F G.f.: G=G(t,z) satisfies tG^3 + tG^2 - z(1+2t)G + z^2*(1+t) = 0.

%e Triangle starts:

%e   1;

%e   0,  3,  1;

%e   0,  0, 12,  9,  2;

%e   0,  0,  0, 55, 66, 30,  5;

%p T:=(n,k)->binomial(3*n-3,n+k)*binomial(k-1,k-n+1)/(n-1): for n from 2 to 10 do seq(T(n,k),k=1..2*n-3) od; # yields sequence in triangular form

%t T[n_, k_] := Binomial[3n - 3, n + k] Binomial[k - 1, k - n + 1]/(n - 1);

%t Table[T[n, k], {n, 2, 10}, {k, 1, 2n - 3}] // Flatten (* _Jean-François Alcover_, Jul 29 2018 *)

%Y Cf. A000108, A001764, A045741, A065065, A089434.

%K nonn,tabf

%O 2,3

%A _Emeric Deutsch_, Jan 24 2007

%E Keyword tabl changed to tabf by _Michel Marcus_, Apr 09 2013