A106522 A Pascal type matrix based on the tribonacci numbers.

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 13, 15, 15, 11, 5, 1, 24, 28, 30, 26, 16, 6, 1, 44, 52, 58, 56, 42, 22, 7, 1, 81, 96, 110, 114, 98, 64, 29, 8, 1, 149, 177, 206, 224, 212, 162, 93, 37, 9, 1, 274, 326, 383, 430, 436, 374, 255, 130, 46, 10, 1, 504, 600, 709, 813, 866, 810, 629, 385, 176, 56, 11, 1

Offset

0,4

Comments

Row sums of A106522 mod 2 are A106524.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Riordan array (1/(1-x-x^2-x^3), x/(1-x)).

Number triangle T(n, 0) = A000073(n+2), T(n, k) = T(n-1, k-1) + T(n-1, k).

Sum_{k=0..n} T(n,k) = A001590(n+3).

Sum_{k=0..floor(n/2)} T(n-k, k) = A106523(n).

Example

Triangle begins:
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   7,  8,  7,  4, 1;
  13, 13, 15, 11, 5, 1;

Mathematica

b[n_]:= b[n]= If[n<2, 0, If[n==2, 1, b[n-1] +b[n-2] +b[n-3]]]; (* A000073 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, b[n+2], T[n-1, k-1] +T[n-1, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 06 2021 *)

Prog

(Sage)
@CachedFunction
def b(n): return 0 if (n<2) else 1 if (n==2) else b(n-1) +b(n-2) +b(n-3)
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return b(n+2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 06 2021

Crossrefs

Cf. A000073, A001590 (row sums), A106523 (diagonal sums).

Sequence in context: A107356 A329854 A124725 * A128175 A104040 A338131

Adjacent sequences: A106519 A106520 A106521 * A106523 A106524 A106525

Keyword

easy,nonn,tabl

Author

Paul Barry, May 06 2005

Status

approved