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A258222 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals. 5
1, 1, 1, 1, 2, 2, 1, 3, 10, 5, 1, 4, 24, 74, 14, 1, 5, 44, 297, 706, 42, 1, 6, 70, 764, 4896, 8162, 132, 1, 7, 102, 1565, 17924, 100278, 110410, 429, 1, 8, 140, 2790, 47650, 527844, 2450304, 1708394, 1430, 1, 9, 184, 4529, 104454, 1831250, 18685164, 69533397, 29752066, 4862 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, Lattice path

FORMULA

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258223(n,i).

EXAMPLE

Square array A(n,k) begins:

:  1,    1,      1,      1,       1,       1, ...

:  1,    2,      3,      4,       5,       6, ...

:  2,   10,     24,     44,      70,     102, ...

:  5,   74,    297,    764,    1565,    2790, ...

: 14,  706,   4896,  17924,   47650,  104454, ...

: 42, 8162, 100278, 527844, 1831250, 4953222, ...

MAPLE

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,

      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)

                 + b(x-1, y+1, true, k)  ))

    end:

A:= (n, k)-> b(2*n, 0, false, k):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];

A [n_, k_] := b[2*n, 0, False, k];

Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Apr 23 2016, translated from Maple *)

CROSSREFS

Columns k=0-1 give: A000108, A000698(n+1).

Rows n=0-2 give: A000012, A000027(k+1), A049450(k+1).

Main diagonal gives A292694.

Cf. A258219, A258223.

Sequence in context: A262157 A090447 A241186 * A112324 A061531 A214722

Adjacent sequences:  A258219 A258220 A258221 * A258223 A258224 A258225

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 23 2015

STATUS

approved

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Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)