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A258219 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (x_p+k*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. 7
1, 1, 1, 1, 2, 4, 1, 3, 10, 25, 1, 4, 18, 74, 208, 1, 5, 28, 153, 706, 2146, 1, 6, 40, 268, 1638, 8162, 26368, 1, 7, 54, 425, 3172, 20898, 110410, 375733, 1, 8, 70, 630, 5500, 44164, 307908, 1708394, 6092032, 1, 9, 88, 889, 8838, 82850, 702844, 5134293, 29752066, 110769550 (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, Feynman diagram

Wikipedia, Lattice path

FORMULA

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258220(n,i).

EXAMPLE

Square array A(n,k) begins:

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

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

:    4,   10,    18,    28,    40,     54, ...

:   25,   74,   153,   268,   425,    630, ...

:  208,  706,  1638,  3172,  5500,   8838, ...

: 2146, 8162, 20898, 44164, 82850, 143046, ...

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, (x+k*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, (x+k*y)/y, 1] + b[x-1, y+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Jan 09 2016, after Alois P. Heinz *)

CROSSREFS

Columns k=0-2 give: A005411 (for n>0), A000698(n+1), A005412(n+1).

Rows n=0-2 give: A000012, A000027(k+1), A028552(k+1).

Main diagonal gives A292693.

Cf. A258220, A258222.

Sequence in context: A121426 A190183 A004515 * A036560 A117297 A112973

Adjacent sequences:  A258216 A258217 A258218 * A258220 A258221 A258222

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 23 2015

STATUS

approved

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)