A123603 Triangle T(n,k), 0<=k<=n, read by rows, with T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k-1) + T(n-2,k).

1, 1, 1, 2, 1, 2, 3, 3, 3, 3, 5, 5, 9, 5, 5, 8, 10, 17, 17, 10, 8, 13, 18, 36, 35, 36, 18, 13, 21, 33, 69, 81, 81, 69, 33, 21, 34, 59, 133, 167, 199, 167, 133, 59, 34, 55, 105, 249, 345, 435, 435, 345, 249, 105, 55, 89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89

Offset: 0

Author

Philippe Deléham, Nov 14 2006, Mar 14 2014

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1;
1, 1;
2, 1, 2;
3, 3, 3, 3;
5, 5, 9, 5, 5;
8, 10, 17, 17, 10, 8;
13, 18, 36, 35, 36, 18, 13;
21, 33, 69, 81, 81, 69, 33, 21;
34, 59, 133, 167, 199, 167, 133, 59, 34;
55, 105, 249, 345, 435, 435, 345, 249, 105, 55;
89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89; ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000045, A000129, A322239 (central terms).

Programs

Mathematica

CoefficientList[CoefficientList[Series[1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)
T[0, 0] := 1; T[n_, k_] := If[k < 0 || k > n, 0, T[n - 1, k - 1] + T[n - 1, k] + T[n - 2, k - 2] - T[n - 2, k - 1] + T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Oct 16 2017 *)

Formula

T(n,k) = T(n,n-k).

T(n,0) = Fibonacci(n+1) = A000045(n+1).

T(n+1,1) = A010049(n+1).

Sum_{k,0<=k<=n} T(n,k)*x^k = A000045(n+1), A000129(n+1), A030195(n+1), A015532(n+1) for x = 0, 1, 2, 3 respectively.

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