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

%I A232969 #20 Sep 21 2021 11:51:03
%S A232969 1,6,60,675,7992,97416,1209951,15227190,193507056,2477564820,
%T A232969 31910429520,412987306320,5366341375695,69965422235442,
%U A232969 914825583252396,11991475839917115,157524763370404320,2073261181622482080,27333449595845251524,360903785815145617992
%N A232969 The sequence S(n,n) that enumerates a certain class of lattice paths from (0,0) to (n,n).
%C A232969 See Dziemianczuk for precise definition.
%H A232969 Alois P. Heinz, <a href="/A232969/b232969.txt">Table of n, a(n) for n = 0..884</a> (first 101 terms from Lars Blomberg)
%H A232969 M. Dziemianczuk, <a href="http://paperity.org/p/34654227/counting-lattice-paths-with-four-types-of-steps">Counting Lattice Paths With Four Types of Steps</a>, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.
%p A232969 b:= proc(x, y) option remember; `if`([x, y]=[0$2], 1,
%p A232969       `if`(x>0, add(b(x-1, y+j), j=-1..1), 0)+
%p A232969       `if`(y>0, b(x, y-1), 0)+`if`(y<0, b(x, y+1), 0))
%p A232969     end:
%p A232969 a:= n-> b(n$2):
%p A232969 seq(a(n), n=0..22);  # _Alois P. Heinz_, Sep 21 2021
%t A232969 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 *)
%o A232969 (PARI) \\ Dziemianczuk, Proposition 1
%o A232969 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)));
%o A232969 vector(20,x,x--;S(x,x)) \\ _Lars Blomberg_, Jul 20 2017
%Y A232969 Leading column of array in A232973.
%K A232969 nonn
%O A232969 0,2
%A A232969 _N. J. A. Sloane_, Dec 05 2013
%E A232969 a(8)-a(19) from _Lars Blomberg_, Jul 20 2017