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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,.... 1
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=2..65.

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-Fran├žois Alcover, Jul 29 2018 *)

CROSSREFS

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

Sequence in context: A144402 A264429 A324163 * A265314 A307791 A307766

Adjacent sequences:  A127534 A127535 A127536 * A127538 A127539 A127540

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Jan 24 2007

EXTENSIONS

Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013

STATUS

approved

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Last modified September 15 18:47 EDT 2019. Contains 327083 sequences. (Running on oeis4.)