A390369 Triangle read by rows: T(n,0) = T(n,n) = 0 and T(n,k) = T(n, k-1) + T(n-1, k-1) + T(n-2, k-1) + 1 for 0 < k < n.

0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 4, 8, 0, 0, 1, 4, 12, 21, 0, 0, 1, 4, 13, 34, 56, 0, 0, 1, 4, 13, 39, 95, 152, 0, 0, 1, 4, 13, 40, 114, 266, 419, 0, 0, 1, 4, 13, 40, 120, 330, 749, 1169, 0, 0, 1, 4, 13, 40, 121, 356, 953, 2122, 3292, 0, 0, 1, 4, 13, 40, 121, 363, 1050, 2753, 6045, 9338, 0

Offset

0,9

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Formula

T(n+1, n) = A383527(n) - 1.

T(n+2, n) = A005773(n+1) - 1.

T(n+2, floor(n/2)+1) = A052993(n).

Sum_{k=0..n+1} T(2*n+2, k) = A000340(n).

For n >= 2*k, T(n, k) = (3^k - 1)/2 = A003462(k).

Example

Triangle T(n, k) starts:
  n\k :     0     1     2     3     4     5     6     7     8     9
  =================================================================
   0  :     0
   1  :     0     0
   2  :     0     1     0
   3  :     0     1     3     0
   4  :     0     1     4     8     0
   5  :     0     1     4    12    21     0
   6  :     0     1     4    13    34    56     0
   7  :     0     1     4    13    39    95   152     0
   8  :     0     1     4    13    40   114   266   419     0
   9  :     0     1     4    13    40   120   330   749  1169     0
  ...

Maple

T := proc (n, k) option remember; if k = 0 or k = n then 0 else T(n, k-1)+T(n-1, k-1)+T(n-2, k-1) +1 end if end proc: seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);

Mathematica

A390369[n_, k_] := A390369[n, k] = If[k == 0 || k == n, 0, A390369[n, k-1] + A390369[n-1, k-1] + A390369[n-2, k-1] + 1];
Table[A390369[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 03 2025 *)

Crossrefs

Cf. A000340, A003462 (central terms), A005773, A052993, A247287 (row sums), A383527, A389359.

Sequence in context: A300725 A035630 A126723 * A325846 A325735 A235794

Adjacent sequences: A390366 A390367 A390368 * A390370 A390371 A390372

Keyword

nonn,easy,tabl

Author

Mélika Tebni, Nov 03 2025

Status

approved