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A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k. 3
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; text; internal format)
OFFSET

2,4

COMMENTS

A Riordan triangle.

Subtriangle of triangle in A167772. [Philippe Deléham, 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. [Philippe Deléham, Jan 23 2010]

T(n,k) = A167772(n-1,k-1), k=2..n. - Reinhard Zumkeller, May 15 2014

LINKS

Reinhard Zumkeller, Rows n=2..125 of triangle, flattened

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

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, Feb 23 2004

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.

Triangle begins:

  1;

  1,1;

  3,2,1;

  8,6,3,1;

  ...

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);

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}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)

PROG

(Haskell)

a065602 n k = sum

   [(k-1+2*j) * a007318' (2*n-k-1-2*j) (n-1) `div` (2*n-k-1-2*j) |

    j <- [0 .. div (n-k) 2]]

a065602_row n = map (a065602 n) [2..n]

a065602_tabl = map a065602_row [2..]

-- Reinhard Zumkeller, May 15 2014

CROSSREFS

Row sums give A000957 (the Fine sequence). First column is A000958.

Cf. A007318.

Sequence in context: A193924 A110439 A327917 * A237596 A292898 A198498

Adjacent sequences:  A065599 A065600 A065601 * A065603 A065604 A065605

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane, Dec 02 2001

EXTENSIONS

More terms from Emeric Deutsch, Feb 23 2004

STATUS

approved

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Last modified January 20 11:01 EST 2020. Contains 331083 sequences. (Running on oeis4.)