A194564 Triangle read by rows: number of reflection-invariant dissections of an n-gon with k noncrossing diagonals.

1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 4, 2, 2, 1, 2, 4, 2, 2, 0, 1, 3, 9, 9, 13, 5, 5, 1, 3, 9, 9, 13, 5, 5, 0, 1, 4, 16, 24, 46, 34, 44, 14, 14, 1, 4, 16, 24, 46, 34, 44, 14, 14, 0, 1, 5, 25, 50, 120, 130, 208, 126, 154, 42, 42, 1, 5, 25, 50, 120, 130, 208, 126, 154, 42, 42, 0

Offset

3,12

Comments

For n odd, the line of reflection passes through a vertex and the midpoint of its opposite edge. For n even, the line of reflection passes through the midpoints of two opposite edges.

References

Ronald C. Read, On general dissections of a polygon, Aequationes Mathematicae, 18 (1978), 370-388.

Links

Table of n, a(n) for n=3..80.

Formula

G.f.: T(x,y) = (y + R(x,y))/(1 - R(x,y)), where

R(x,y) = x*V(x^2, y^2)/(1 - V(x^2, y^2)) and

V(x,y) = y + sum(k>=0, sum(n>=k+2, x^k*y^n*(1/(k+1))*C(n-3, k)*C(n+k-1, n-1) ) ).

Example

The triangle begins:
n=3:   1
n=4:   1,0
n=5:   1,1,1
n=6:   1,1,1,0
n=7:   1,2,4,2,2
n=8:   1,2,4,2,2,0
n=9:   1,3,9,9,13,5,5
n=10:  1,3,9,9,13,5,5,0
n=11:  1,4,16,24,46,34,44,14,14
n=12:  1,4,16,24,46,34,44,14,14,0
n=13:  1,5,25,50,120,130,208,126,154,42,42
n=14:  1,5,25,50,120,130,208,126,154,42,42,0
n=15:  1,6,36,90,260,370,708,622,902,468,552,132,132
n=16:  1,6,36,90,260,370,708,622,902,468,552,132,132,0

Crossrefs

Sequence in context: A123330 A300821 A368795 * A284690 A064132 A072865

Adjacent sequences: A194561 A194562 A194563 * A194565 A194566 A194567

Keyword

nonn,tabl

Author

Daniel Hess, Aug 28 2011

Status

approved