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 A339390 Number of paths from (0,0,0) to (n,n,n) using steps (1,0,0), (0,1,0), (0,0,1), (1,1,1), and (2,2,2). 2
 1, 7, 116, 2397, 54845, 1329644, 33464881, 864627351, 22776683200, 609024723535, 16478750543705, 450190397799036, 12397538372467109, 343712858468053319, 9584085091610235280, 268571959802603851989, 7558772037473679862681, 213548821612723752662596 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The ratio of any two consecutive terms of this sequence a(n+1)/a(n) seems to grow asymptotically to ~30 as n increases (observation). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..679 FORMULA From Alois P. Heinz, Dec 05 2020: (Start) a(n) = [(x*y*z)^n] 1/(1-x-y-z-x*y*z-(x*y*z)^2). a(n) = ((3*n-7)*(3*n-2)*(30*n^2-50*n+13)*a(n-1) - (3*n-2)*(3*n-5)*a(n-2) - (45*n^4-300*n^3+677*n^2-560*n+108)*a(n-3) + (3*n-2)*(3*n-11)*a(n-4) + (3*n-1)*(9*n^3-75*n^2+197*n-154)*a(n-5) + (3*n-1)*(3*n-4)*(n-4)^2*a(n-6)) / ((3*n-4)*(3*n-7)*n^2) for n>=6. (End) MAPLE b:= proc(l) option remember; `if`(l[3]=0, 1, add((f-> `if`(f[1]<0, 0, b(f)))(sort(l-h)), h= [[1, 0\$2], [0, 1, 0], [0\$2, 1], [1\$3], [2\$3]])) end: a:= n-> b([n\$3]): seq(a(n), n=0..20); # Alois P. Heinz, Dec 04 2020 # second Maple program: a:= proc(n) local t; 1/(1-x-y-z-x*y*z-(x*y*z)^2); for t in [x, y, z] do coeftayl(%, t=0, n) od end: seq(a(n), n=0..20); # Alois P. Heinz, Dec 05 2020 # third Maple program: a:= proc(n) option remember; `if`(n<6, [1, 7, 116, 2397, 54845, 1329644][n+1], ((3*n-7)*(3*n-2)*(30*n^2-50*n+13)*a(n-1) -(3*n-2) *(3*n-5)*a(n-2) -(45*n^4-300*n^3+677*n^2-560*n+108)*a(n-3) +(3*n-2)*(3*n-11)*a(n-4) +(3*n-1)*(9*n^3-75*n^2+197*n-154)*a(n-5) +(3*n-1)*(3*n-4)*(n-4)^2*a(n-6)) / ((3*n-4)*(3*n-7)*n^2)) end: seq(a(n), n=0..20); # Alois P. Heinz, Dec 05 2020 MATHEMATICA b[l_] := b[l] = If[l[[3]] == 0, 1, Sum[Function[f, If[f[[1]] < 0, 0, b[f]]][Sort[l-h]], {h, {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 1}, {2, 2, 2}}}]]; a[n_] := b[{n, n, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *) CROSSREFS Cf. A006480, A081798, A126086, A268550, A339565. Sequence in context: A328813 A178297 A285394 * A251585 A320083 A329543 Adjacent sequences: A339387 A339388 A339389 * A339391 A339392 A339393 KEYWORD nonn AUTHOR William J. Wang, Dec 02 2020 STATUS approved

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Last modified November 30 02:32 EST 2022. Contains 358431 sequences. (Running on oeis4.)