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A232969
The sequence S(n,n) that enumerates a certain class of lattice paths from (0,0) to (n,n).
2
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 05 2013
EXTENSIONS
a(8)-a(19) from Lars Blomberg, Jul 20 2017
STATUS
approved