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A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).

%I A339565 #29 Mar 26 2025 17:32:22
%S A339565 1,3,17,101,627,3999,25955,170571,1131433,7559301,50795985,342935689,
%T A339565 2324278669,15804931797,107775401349,736723618773,5046774983235,
%U A339565 34636814325087,238114193665451,1639378334244867,11301978856210543,78010917772099207,539055832175992119
%N A339565 Number of lattice paths from (0,0) to (n,n) using steps (0,1), (1,0), (1,1), (1,2), (2,1).
%H A339565 Alois P. Heinz, <a href="/A339565/b339565.txt">Table of n, a(n) for n = 0..1179</a> (first 101 terms from Kent Mei)
%F A339565 a(n) = [(x*y)^n] 1/(1-x-y-x*y-x*y^2-x^2*y). - _Alois P. Heinz_, Dec 09 2020
%F A339565 a(n) = A382436(2n,n). - _Alois P. Heinz_, Mar 25 2025
%F A339565 a(n) ~ sqrt((3776 + (26570110976 - 74946048*sqrt(177))^(1/3) + 8*(59*(879572 + 2481*sqrt(177)))^(1/3))/11328) * (2 + (459 - 12*sqrt(177))^(1/3)/3 + (153 + 4*sqrt(177))^(1/3)/3^(2/3))^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 26 2025
%p A339565 a:= proc(n) local t; 1/(1-x-y-x*y-(x*y^2)-(x^2*y));
%p A339565       for t in [x, y] do coeftayl(%, t=0, n) od
%p A339565     end:
%p A339565 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 09 2020
%p A339565 # second Maple program:
%p A339565 b:= proc(l) option remember; `if`(l[2]=0, 1,
%p A339565       add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h=
%p A339565       [[1, 0], [0, 1], [1$2], [1, 2], [2, 1]]))
%p A339565     end:
%p A339565 a:= n-> b([n$2]):
%p A339565 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 09 2020
%p A339565 # third Maple program:
%p A339565 a:= proc(n) option remember; `if`(n<3, [1, 3, 17][n+1],
%p A339565       ((6*n-3)*a(n-1)+(7*n-7)*a(n-2)+(4*n-6)*a(n-3))/n)
%p A339565     end:
%p A339565 seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 09 2020
%t A339565 b[l_] := b[l] = If[l[[2]] == 0, 1,
%t A339565      Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l - h]], {h,
%t A339565      {{1, 0}, {0, 1}, {1, 1}, {1, 2}, {2, 1}}}]];
%t A339565 a[n_] := b[{n, n}];
%t A339565 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 30 2022, after _Alois P. Heinz_ *)
%Y A339565 Cf. A000984, A001850, A137635, A339390, A382436.
%K A339565 nonn
%O A339565 0,2
%A A339565 _Kent Mei_, Dec 08 2020