A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 5, 3, 1, 10, 11, 10, 8, 4, 1, 22, 21, 21, 18, 12, 5, 1, 42, 43, 42, 39, 30, 17, 6, 1, 86, 85, 85, 81, 69, 47, 23, 7, 1, 170, 171, 170, 166, 150, 116, 70, 30, 8, 1, 342, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 682, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Author

Paul Barry, Jan 23 2004

Keywords

• easy
• nonn
• tabl

Comments

A Jacobsthal-Pascal triangle.

Examples

			Triangle starts as:
   1;
   0,  1;
   2,  1,  1;
   2,  3,  2,  1;
   6,  5,  5,  3,  1;
  10, 11, 10,  8,  4,  1;
  22, 21, 21, 18, 12,  5, 1;
  42, 43, 42, 39, 30, 17, 6, 1; ...
		

Links

• G. C. Greubel, Rows n = 0..20 of triangle, flattened

Crossrefs

Columns include A078008, A001045, A000975, A011377. Row sums give A084219.

Cf. A091597.

Programs

Mathematica

T[n_, k_]:= If[k==0, (2^n + 2*(-1)^n)/3, If[k<0 || k>n, 0, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

PARI

{T(n,k) = if(k==0, (2^n + 2*(-1)^n)/3, if(k<0 || k>n, 0, T(n-1,k-1) + T(n-1,k)))}; \\ G. C. Greubel, Jun 04 2019

Sage

def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return (2^n + 2*(-1)^n)/3
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jun 04 2019

Formula

k-th column has e.g.f. ((1-x)/(1-x-x^2))*(x/(1-x))^k.