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A120059
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Triangle read by rows: T(n,k) is the number of Dyck n-paths (A000108) whose longest pyramid has size k.
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1
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1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 2, 1, 0, 13, 19, 7, 2, 1, 0, 35, 63, 24, 7, 2, 1, 0, 97, 212, 85, 25, 7, 2, 1, 0, 275, 723, 307, 90, 25, 7, 2, 1, 0, 794, 2491, 1121, 330, 91, 25, 7, 2, 1, 0, 2327, 8654, 4129, 1225, 335, 91, 25, 7, 2, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| A pyramid in a Dyck path is a subpath of the form U^k D^k with k>=1 (U=upstep, D=downstep). The longest pyramid is indicated by lowercase letters in the Dyck path UUDuuudddDUD and it has size 3.
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FORMULA
| Generating function for column k>=1 is F[k]-F[k-1] where F[k]:=(1 + x^(k+1) - ((1 + x^(k+1))^2 - 4*x)^(1/2))/(2*x).
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EXAMPLE
| Table begins
\ k..0....1....2....3....4....5....6....7
n
0 |..1
1 |..0....1
2 |..0....1....1
3 |..0....2....2....1
4 |..0....5....6....2....1
5 |..0...13...19....7....2....1
6 |..0...35...63...24....7....2....1
7 |..0...97..212...85...25....7....2....1
a(3,2)=2 because the Dyck 3-paths whose longest pyramid has size 2 are
UUDDUD, UDUUDD.
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MATHEMATICA
| Clear[a] (* a[n, k] is the number of Dyck n-paths whose longest pyramid has size <=k *) a[0, k_]/; k>=0 := 1 a[1, k_]/; k>=1 := 1 a[n_, k_]/; k>=n := 1/(n+1)Binomial[2n, n] a[n_, 0]/; n>=1 := 0 a[n_, k_]/; k<0:= 0 a[n_, k_]/; 1<=k && k<n && n>=2 := a[n, k] = Sum[a[j-1, k] a[n-j, k], {j, n}] - a[n-k-1, k] Table[a[n, k]-a[n, k-1], {n, 0, 8}, {k, 0, n}]
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CROSSREFS
| Cf. A120060. Column k=1 is A086581. Row sums are the Catalan numbers A000108.
Sequence in context: A205574 A049244 A110281 * A067347 A120568 A189233
Adjacent sequences: A120056 A120057 A120058 * A120060 A120061 A120062
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KEYWORD
| nonn,tabl
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AUTHOR
| David Callan (callan(AT)stat.wisc.edu), Jun 06 2006
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