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Search: id:A098747
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| A098747 |
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level. |
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+0
1
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| 1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 24, 11, 5, 1, 1, 75, 35, 14, 6, 1, 1, 243, 113, 47, 17, 7, 1, 1, 808, 376, 156, 60, 20, 8, 1, 1, 2742, 1276, 532, 204, 74, 23, 9, 1, 1, 9458, 4402, 1840, 712, 257, 89, 26, 10, 1, 1, 33062, 15390, 6448, 2507, 917, 315, 105, 29, 11, 1, 1, 116868
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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T(n,0)=A000958(n-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2006
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REFERENCES
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Y. Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 237 (2004), 177-186.
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FORMULA
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See Mathematica line.
G.f.=zC/(1+z-tz-zC), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2006
With offset 0 (0<=k<=n), T(n,k)=A065600(n,k)+A065600(n+1,k)-A065600(n,k-1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 01 2007
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EXAMPLE
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Triangle begins:
1
1 1
3 1 1
8 4 1 1
24 11 5 1 1
75 35 14 6 1 1
T(4,2)=1 because we have UDUDUUDD.
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MAPLE
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c:=(1-sqrt(1-4*z))/2/z: G:=z*c/(1-t*z+z-z*c): Gser:=simplify(series(G, z=0, 15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 12 do seq(coeff(P[n], t, k), k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 23 2006
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MATHEMATICA
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u[n_, k_, i_]:=(2i+1)/(n-k)Binomial[k+i, i]Binomial[2n-2k-2i-2, n-k-1] u[n_, k_]/; k<=n-1 := Sum[u[n, k, i], {i, 0, n-k-1}] Table[u[n, k], {n, 10}, {k, 0, n-1}] (* u[n, k, i] is the number of Dyck n-paths with k low UDUs and k+i+1 returns altogether. For example, with n=4, k=1, and i=1, u[n, k, i] counts UDUUDDUD, UUDDUDUD because each has size n=4, k=1 low UDUs, and k+i+1=3 returns to ground level. *) (David Callan, Nov 03 2005)
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CROSSREFS
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Cf. A091869, A092107.
Cf. A000958.
Sequence in context: A016462 A121461 A114276 this_sequence A122897 A117425 A091698
Adjacent sequences: A098744 A098745 A098746 this_sequence A098748 A098749 A098750
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KEYWORD
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nonn,tabl
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AUTHOR
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njas, Oct 30 2004
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