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A116424
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Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UDUU's, 0 <= k <= floor((n-1)/2).
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3
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1, 1, 2, 4, 1, 9, 5, 22, 19, 1, 57, 66, 9, 154, 221, 53, 1, 429, 729, 258, 14, 1223, 2391, 1131, 116, 1, 3550, 7829, 4652, 745, 20, 10455, 25638, 18357, 4115, 220, 1, 31160, 84033, 70404, 20598, 1790, 27, 93802, 275765, 264563, 96286, 12104, 379, 1, 284789
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OFFSET
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0,3
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COMMENTS
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T(n,k) also gives the number of Dyck paths of semilength n with k UUDU's.
Column k=0 gives A105633(n-1) for n > 0.
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LINKS
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FORMULA
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T(n,k) = Sum_{i=k..floor((n-1)/2)} (-1)^(i+k) * binomial(i,k) * binomial(n-i,i) * binomial(2*n-3*i, n - 2*i -1)/(n-i), n >= 1.
G.f. G = G(t,z) satisfies G = 1 + z^2(1-t)G + z(1-z+tz)G^2.
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EXAMPLE
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Triangle begins:
00 : 1;
01 : 1;
02 : 2;
03 : 4, 1;
04 : 9, 5;
05 : 22, 19, 1;
06 : 57, 66, 9;
07 : 154, 221, 53, 1;
08 : 429, 729, 258, 14;
09 : 1223, 2391, 1131, 116, 1;
10 : 3550, 7829, 4652, 745, 20;
...
T(4,1) = 5 because there exist five Dyck paths of semilength 4 with one occurrence of UDUU : UDUUUDDD, UDUUDUDD, UDUUDDUD, UUDUUDDD, UDUDUUDD.
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MAPLE
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b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])*
`if`(t=4, z, 1) +b(x-1, y-1, [1, 3, 1, 3][t]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
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MATHEMATICA
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s = Series[((1 + (t - 1) z^2) - Sqrt[(1 + (t - 1) z^2)^2 - 4*z*(1 - z + z*t)])/(2*z*(1 - z + z*t)), {z, 0, 15}] // CoefficientList[#, z]&;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
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STATUS
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approved
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