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 A218321 Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k>=0. 4
 1, 2, 8, 39, 212, 1230, 7458, 46689, 299463, 1957723, 12996879, 87383754, 593794311, 4071599216, 28136612051, 195756911831, 1370068168916, 9639404836227, 68138551870047, 483682445360748, 3446462104490724, 24642148415136556, 176743014104068411 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 Alois P. Heinz, Maple program for A218321 FORMULA G.f.: (sqrt(x^4+4*x^3+2*x^2-8*x+1)+x^2+1-sqrt(2*(x^4+2*x^3-6*x^2-4*x+1+(x^2+1)*sqrt(x^4+4*x^3+2*x^2-8*x+1))))/(4*x^2). - Mark van Hoeij, Apr 17 2013 EXAMPLE a(2) = 8: [(0,0),(1,0),(1,1),(2,1),(2,2)], [(0,0),(1,0),(1,1),(2,2)], [(0,0),(1,0),(2,0),(2,1),(2,2)], [(0,0),(1,0),(2,1),(2,2)], [(0,0),(1,0),(2,2)], [(0,0),(1,1),(2,1),(2,2)], [(0,0),(1,1),(2,2)], [(0,0),(2,1),(2,2)]. MAPLE b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1, add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y) -b(x-1, y-1))) end: a:= n-> b(n, n): seq(a(n), n=0..30); # second Maple program gives series: series(RootOf(x^4*T^4-(x^2+1)*x^2*T^3-(x^2-2*x-2)*x*T^2-(x^2+1)*T+1, T), x=0, 31); # Mark van Hoeij, Apr 17 2013 MATHEMATICA b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - i, y - 1], {i, 0, x}] + Sum[b[x - 1, y - j], {j, 0, y}] - b[x - 1, y - 1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *) CROSSREFS Cf. A082582, A168592, A263316. Sequence in context: A206901 A162476 A366049 * A236339 A292100 A185650 Adjacent sequences: A218318 A218319 A218320 * A218322 A218323 A218324 KEYWORD nonn AUTHOR Alois P. Heinz, Oct 25 2012 STATUS approved

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Last modified September 12 15:00 EDT 2024. Contains 375851 sequences. (Running on oeis4.)