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A341415
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Triangle read by rows: T(n,k) is the number of grand Dyck paths of semilength n having degree of symmetry k (n >= 0, 0 <= k <= n).
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0
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1, 0, 2, 2, 0, 4, 4, 8, 0, 8, 14, 16, 24, 0, 16, 44, 64, 48, 64, 0, 32, 148, 208, 216, 128, 160, 0, 64, 504, 736, 720, 640, 320, 384, 0, 128, 1750, 2592, 2672, 2176, 1760, 768, 896, 0, 256, 6156, 9280, 9696, 8448, 6080, 4608, 1792, 2048, 0, 512
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OFFSET
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0,3
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COMMENTS
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The degree of symmetry of a grand Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along a vertical line through the midpoint of the path.
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LINKS
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FORMULA
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G.f.: 1/(2(1-u)z+sqrt(1-4z)).
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EXAMPLE
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For n=3 there are 4 grand Dyck paths with degree of symmetry equal to 0, namely uddduu, uudddu, duuudd, dduuud.
The triangle begins:
1
0 2
2 0 4
4 8 0 8
14 16 24 0 16
44 64 48 64 0 32
148 208 216 128 160 0 64
504 736 720 640 320 384 0 128
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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