A339565 -id:A339565 - cp's OEIS frontend

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 9, 17, 9, 1, 1, 12, 36, 36, 12, 1, 1, 15, 64, 101, 64, 15, 1, 1, 18, 101, 227, 227, 101, 18, 1, 1, 21, 147, 440, 627, 440, 147, 21, 1, 1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1, 1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1

Offset: 0

F. Chapoton, Mar 25 2025

The original definition was "Decomposition of A077938".

Every row is symmetric.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   6,    1;
  1,  9,  17,    9,    1;
  1, 12,  36,   36,   12,    1;
  1, 15,  64,  101,   64,   15,    1;
  1, 18, 101,  227,  227,  101,   18,    1;
  1, 21, 147,  440,  627,  440,  147,   21,   1;
  1, 24, 202,  767, 1459, 1459,  767,  202,  24,  1;
  1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1;
  ...

Similar to A008288, A103450, and A382444.
Row sums are A077938.
T(2n, n) gives A339565.
Cf. A056594.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
inverse = 1 + (-y - 1)*x - y*x^2 + (-y^2 - y)*x^3
gf = 1 / inverse
[list(u) for u in list(gf.O(11))]

G.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

Sum_{k=0..n} (-1)^k * T(n,k) = A056594(n). - Alois P. Heinz, Mar 25 2025

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).

Original entry on oeis.org

1, 7, 116, 2397, 54845, 1329644, 33464881, 864627351, 22776683200, 609024723535, 16478750543705, 450190397799036, 12397538372467109, 343712858468053319, 9584085091610235280, 268571959802603851989, 7558772037473679862681, 213548821612723752662596

Offset: 0

William J. Wang, Dec 02 2020

nonn

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).

Alois P. Heinz, Table of n, a(n) for n = 0..679

Cf. A006480, A081798, A126086, A268550, A339565.

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

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 *)

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)

cp's OEIS Frontend

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

Formula

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).