OFFSET
1,2
LINKS
D. Birmajer, J. B. Gil, and M. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.
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).
EXAMPLE
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.
MATHEMATICA
Rest[CoefficientList[(InverseSeries[Series[(y-2*y^2+y^5-y^6)/(1-y), {y, 0, 24}], x]-x)/x, x]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael D. Weiner, Apr 09 2015
STATUS
approved