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A065602
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Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.
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1
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1, 1, 1, 3, 2, 1, 8, 6, 3, 1, 24, 18, 10, 4, 1, 75, 57, 33, 15, 5, 1, 243, 186, 111, 54, 21, 6, 1, 808, 622, 379, 193, 82, 28, 7, 1, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1, 33062, 25724, 16266, 8928, 4332, 1856, 692, 218
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,4
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COMMENTS
| A Riordan triangle.
Subtriangle of triangle in A167772. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2009]
Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A000958 and g(x) is the g.f. of A000108. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 23 2010]
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REFERENCES
| E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
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FORMULA
| T(n, k)= sum((k-1+2j)*binomial(2n-k-1-2j, n-1)/(2n-k-1-2j), j=0..floor((n-k)/2)). G.f.=t^2*z^2*C/[(1-z^2*C^2)(1-tzC)], where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
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EXAMPLE
| Example: T(3,2)=1 reflecting the unique Dyck path (UUDUDD) of length 6, with no hills and height of first peak equal to 2.
1; 1,1; 3,2,1; 8,6,3,1; ...
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MAPLE
| a := proc(n, k) if n=0 and k=0 then 1 elif k<2 or k>n then 0 else sum((k-1+2*j)*binomial(2*n-k-1-2*j, n-1)/(2*n-k-1-2*j), j=0..floor((n-k)/2)) fi end: seq(seq(a(n, k), k=2..n), n=1..14);
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MATHEMATICA
| nmax = 12; t[n_, k_] := Sum[(k-1+2j)*Binomial[2n-k-1-2j, n-1] / (2n-k-1-2j), {j, 0, (n-k)/2}]; Flatten[ Table[t[n, k], {n, 2, nmax}, {k, 2, n}]] (* From Jean-François Alcover, Nov 08 2011, after Maple *)
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CROSSREFS
| Row sums give A000957 (the Fine sequence). First column is A000958.
Sequence in context: A090452 A193924 A110439 * A198498 A016648 A104552
Adjacent sequences: A065599 A065600 A065601 * A065603 A065604 A065605
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 02 2001
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 23 2004
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