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A108198
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Triangle read by rows: T(n,k)=binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0<=k<=n).
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2
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1, 1, 2, 1, 4, 5, 1, 6, 15, 14, 1, 8, 30, 56, 42, 1, 10, 50, 140, 210, 132, 1, 12, 75, 280, 630, 792, 429, 1, 14, 105, 490, 1470, 2772, 3003, 1430, 1, 16, 140, 784, 2940, 7392, 12012, 11440, 4862, 1, 18, 180, 1176, 5292, 16632, 36036, 51480, 43758, 16796, 1, 20, 225
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also, with offset 1, triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and ending at the point (2k,0) (1<=k<=n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. For example, T(3,2)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDDL.
Sum of row n = A002212(n+1). T(n,n)=Catalan(n+1) (A000108).
Sum(k*T(n,k),k=1..n)=A026388(n). Mirror image of A126181.
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| With offset 1, T(n,k)=c(k)*binom(n-1,k-1), where c(j)=binom(2j,j)/(j+1) is a Catalan number (A000108). G.f.=G-1, where G=G(t,z) satisfies G=1+tzG^2+z(G-1)
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EXAMPLE
| Triangle begins:
1;
1,2;
1,4,5;
1,6,15,14;
1,8,30,56,42;
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MAPLE
| T:=(n, k)->binomial(2*k+2, k+1)*binomial(n, k)/(k+2): for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A002212, A000108, A026388, A126181.
Sequence in context: A050166 A124959 A081281 * A121289 A134248 A080935
Adjacent sequences: A108195 A108196 A108197 * A108199 A108200 A108201
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 15 2005, Mar 30 2007
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Pewe, Jun 16 2007
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