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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.)