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A127537
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,....
2
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, 0, 0, 1428, 3060, 2856, 1428, 378, 42, 0, 0, 0, 0, 0, 0, 7752, 20349, 23940, 15960, 6300, 1386, 132, 0, 0, 0, 0, 0, 0, 0, 43263, 134596, 191268, 159390, 83490, 27324, 5148, 429
OFFSET
2,3
COMMENTS
Row n contains 2n-3 terms, the first n-2 of which are equal to 0.
T(n,n-1) = A001764(n-1). T(n,2n-3) = A000108(n-2) (the Catalan numbers).
T(n,k) = A089434(n,k+1-n).
Sum_{k=n-1..2n-3} k*T(n,k) = A045741(n).
Sum_{n=k..2k-2} T(n,k) = A065065(k).
LINKS
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
C. Domb & A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
C. Domb & A. J. Barrett, Notes on Table 2 in "Enumeration of ladder graphs", Discrete Math. 9 (1974), 55. (Annotated scanned copy)
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
FORMULA
T(n,k) = C(3n-3,n+k)C(k-1,k-n+1)/(n-1) (n >= 2, 0 <= k <= 2n-3).
G.f.: G=G(t,z) satisfies tG^3 + tG^2 - z(1+2t)G + z^2*(1+t) = 0.
EXAMPLE
Triangle starts:
1;
0, 3, 1;
0, 0, 12, 9, 2;
0, 0, 0, 55, 66, 30, 5;
MAPLE
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
MATHEMATICA
T[n_, k_] := Binomial[3n - 3, n + k] Binomial[k - 1, k - n + 1]/(n - 1);
Table[T[n, k], {n, 2, 10}, {k, 1, 2n - 3}] // Flatten (* Jean-François Alcover, Jul 29 2018 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jan 24 2007
EXTENSIONS
Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013
STATUS
approved