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A194564
Triangle read by rows: number of reflection-invariant dissections of an n-gon with k noncrossing diagonals.
0
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.
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
KEYWORD
nonn,tabl
AUTHOR
Daniel Hess, Aug 28 2011
STATUS
approved