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A089435
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Triangle read by rows: T(n,k) (n >=2, k >=0) is the number of non-crossing connected graphs on n nodes on a circle, having k triangles. Rows are indexed 2,3,4,...; columns are indexed 0,1,2,....
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0
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1, 3, 1, 13, 8, 2, 66, 60, 25, 5, 367, 442, 255, 84, 14, 2164, 3248, 2380, 1064, 294, 42, 13293, 23904, 21192, 11832, 4410, 1056, 132, 84157, 176397, 183303, 122115, 56430, 18216, 3861, 429, 545270, 1305480, 1554850, 1200320, 657195, 262262, 75075
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math. 204 (1999), 203-229.
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FORMULA
| T(n, k)=binomial(n+k-2, k)*sum(binomial(n+k+i-2, i)*binomial(3*n-3-k-i, 2*n-1+i), i=0..floor((n-k-2)/2))/(n-1), n>=2, k>=0.
G.f.: G(t, z) satisfies G^4+G^3+(t-4)*z*G^2-2*(t-2)*z^2*G+(t-1)*z^3=0.
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EXAMPLE
| T(4,1)=8 because, considering the complete graph K_4 on the nodes A,B,C and D, we obtain a non-crossing connected graph on A,B,C,D, with exactly one triangle, by deleting one of the two diagonals and one of the four sides (8 possibilities).
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MATHEMATICA
| t[n_, k_] = Binomial[n+k-2, k]*Sum[Binomial[n+k+i-2, i]*Binomial[3n-3-k-i, 2n-1+i], {i, 0, Floor[(n-k-2)/2]}]/(n-1) ;
Flatten[Table[t[n, k], {n, 2, 10}, {k, 0, n-2}]][[1 ;; 43]] (* From Jean-François Alcover, Jun 20 2011 *)
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CROSSREFS
| T(n, n-2) yields the Catalan numbers (A000108) corresponding to triangulations, T(n, 0) yields A045743, row sums yield A007297.
Cf. A007297, A000108, A045743.
Sequence in context: A096773 A118384 A133176 * A152474 A088814 A088729
Adjacent sequences: A089432 A089433 A089434 * A089436 A089437 A089438
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2003
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