A256752 Number of ways to place non-intersecting diagonals in a convex (n+2)-gon so as to create no hexagons.
1, 3, 11, 44, 190, 859, 4015, 19248, 94117, 467575, 2353443, 11975568, 61505088, 318406927, 1659801852, 8704865907, 45898065978, 243163198928, 1293769867676, 6910165762943, 37036898772008, 199140325574519, 1073849938338566
Offset: 1
Keywords
Examples
a(3)=11 because all 11 dissections of the pentagon are allowed, i.e., the null placement, 5 placements of 1 diagonal and 5 placements of two diagonals.
Links
- D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.
Programs
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Mathematica
Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^5-y^6)/(1-y), {y, 0, 24}], x]-x)/x, x]]
Formula
a(n) = (1/(n+1))*Sum_{i=0..floor(n/4)} Sum_{k=i+1..n-3*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-4*i-1,k-i-1), n !== 0 (mod 4),
a(n) = ((-1)^(n/4)/(n+1))*binomial(5*n/4,n/4) + (1/(n+1))*Sum_{i=0..(n/4)-1} Sum_{k=i+1..n-3*i} (-1)^i*binomial(n+k,k)*binomial(k,i)*binomial(n-4*i-1,k-i-1), n == 0 (mod 4).