A045744
Number of noncrossing connected graphs on n nodes on a circle having no four-sided faces.
Original entry on oeis.org
1, 4, 22, 141, 988, 7337, 56749, 452332, 3689697, 30652931, 258465558, 2206330790, 19029531220, 165582392070, 1451789520435, 12813638048184, 113755675163767, 1015119850103821, 9100463691522759, 81923222827031025
Offset: 2
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a(n) = if(n>1, sum(i=0, floor((n-2)/3), binomial(n-2+i, i)*binomial(4*n-4-i, n-2-3*i))/(n-1)); \\ Andrew Howroyd, Nov 12 2017
A089435
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,....
Original entry on oeis.org
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
Offset: 2
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).
Triangle starts:
1;
3, 1;
13, 8, 2;
66, 60, 25, 5;
367, 442, 255, 84, 14;
...
T(n, n-2) yields the Catalan numbers (
A000108) corresponding to triangulations, T(n, 0) yields
A045743, row sums yield
A007297.
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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]] (* Jean-François Alcover, Jun 20 2011 *)
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T(n, k) = binomial(n+k-2, k)*sum(i=0,floor((n-k-2)/2),binomial(n+k+i-2, i)*binomial(3*n-3-k-i, 2*n-1+i))/(n-1); \\ Michel Marcus, Oct 26 2015
A366070
Expansion of (1/x) * Series_Reversion( x*(1+x-x^2)/(1+x)^5 ).
Original entry on oeis.org
1, 4, 23, 155, 1143, 8932, 72682, 609348, 5227035, 45659020, 404756300, 3632075109, 32928392154, 301152242600, 2775117150576, 25741623112539, 240162703635495, 2252187478291088, 21217451539791085, 200709823787548845, 1905712342347978340
Offset: 0
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a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n-k+4, n-2*k))/(n+1);
Showing 1-3 of 3 results.