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A118972
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Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having length of first descent equal to k (1<=k<=n; n>=1). A hill in a Dyck path is a peak at level 1.
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3
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0, 0, 1, 1, 0, 1, 3, 2, 0, 1, 10, 5, 2, 0, 1, 33, 16, 5, 2, 0, 1, 111, 51, 16, 5, 2, 0, 1, 379, 168, 51, 16, 5, 2, 0, 1, 1312, 565, 168, 51, 16, 5, 2, 0, 1, 4596, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 16266, 6716, 1934, 565, 168, 51, 16, 5, 2, 0, 1, 58082, 23604, 6716, 1934, 565, 168
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| Row sums are the Fine numbers (A000957). T(n,1)=A001558(n-3) for n>=3. T(n,k)=A118973(n-k) for n>=k>=2. Sum(k*T(n,k),k=1..n)=A118974(n)
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REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241, 241-265, 2001.
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FORMULA
| G:=tz^2*CF[C-(1-t)/(1-tz)], where F=[1-sqrt(1-4z)]/[z(3-sqrt(1-4z)] and C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
| T(5,2)=5 because we have uu(dd)uududd, uu(dd)uuuddd,uuu(dd)uuddd,uuu(dd)ududd and uuuu(dd)uddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
Triangle starts:
0;
0,1;
1,0,1;
3,2,0,1;
10,5,2,0,1;
33,16,5,2,0,1;
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MAPLE
| F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: G:=t*z^2*C*F*(C-(1-t)/(1-t*z)): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000957, A001558, A118973, A118974.
Sequence in context: A142071 A193283 A193277 * A171224 A193233 A145878
Adjacent sequences: A118969 A118970 A118971 * A118973 A118974 A118975
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), May 08 2006
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