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Search: id:A108863
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| A108863 |
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Number of Dyck paths containing exactly one UUUD. |
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+0
1
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| 0, 0, 0, 1, 5, 21, 78, 274, 927, 3061, 9933, 31824, 100972, 317942, 995088, 3099105, 9612735, 29715525, 91595391, 281643480, 864189486, 2646805668, 8093543439, 24713953515, 75370741506, 229604257846, 698754428388, 2124616182139
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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a(n) = number of Dyck n-paths containing exactly one UUUD.
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FORMULA
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G.f. (x-1+(1-2*x)M)/(x(1-3*x)(1+x*M)) = Sum_{n>=0}a(n)x^n where M = (1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2) is the gf for Motzkin numbers (A001006); satisfies z^3 = (1 + z)(1 - 3z)( (1 - 3z + z^2)G + z^2(1 - 3z)G^2 ).
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EXAMPLE
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a(4) = 5 because UUUUDDDD, UUUDUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD
each contain one UUUD.
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CROSSREFS
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Cf. same as A055219 except for offset, and is column k=1 of A091958. Dyck paths containing no UUUD are counted by the Motzkin numbers (A001006).
Adjacent sequences: A108860 A108861 A108862 this_sequence A108864 A108865 A108866
Sequence in context: A134770 A084780 A055219 this_sequence A027172 A029870 A081038
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KEYWORD
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nonn
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AUTHOR
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David Callan (callan(AT)stat.wisc.edu), Jul 25 2005
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