A232969 The sequence S(n,n) that enumerates a certain class of lattice paths from (0,0) to (n,n).

1, 6, 60, 675, 7992, 97416, 1209951, 15227190, 193507056, 2477564820, 31910429520, 412987306320, 5366341375695, 69965422235442, 914825583252396, 11991475839917115, 157524763370404320, 2073261181622482080, 27333449595845251524, 360903785815145617992

Offset

0,2

Comments

See Dziemianczuk for precise definition.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..884 (first 101 terms from Lars Blomberg)

M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.

Maple

b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
      `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
      `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 21 2021

Mathematica

Table[Function[k, Sum[Sum[Binomial[k, j] Binomial[j, i - j] Binomial[2 k + n - i, k], {j, 0, i}], {i, 0, n + k}]]@ n, {n, 0, 19}] (* Michael De Vlieger, Jul 22 2017 *)

Prog

(PARI) \\ Dziemianczuk, Proposition 1
S(n, k)=sum(i=0, n+k, sum(j=0, i, binomial(k, j)*binomial(j, i-j)*binomial(2*k+n-i, k)));
vector(20, x, x--; S(x, x)) \\ Lars Blomberg, Jul 20 2017

Crossrefs

Leading column of array in A232973.

Sequence in context: A106259 A085364 A228484 * A232246 A086984 A000894

Adjacent sequences: A232966 A232967 A232968 * A232970 A232971 A232972

Keyword

nonn

Author

N. J. A. Sloane, Dec 05 2013

Extensions

a(8)-a(19) from Lars Blomberg, Jul 20 2017

Status

approved