A276696 Triangle read by rows, T(n,k) = T(n-1, k-1) + T(n-2, k) if k is odd, T(n-1, k-1) + T(n-1, k) if k is even, for k<=0<=n and n>=2 with T(0,0)=T(1,0)=T(1,1)=0 and T(n,k)=0 when k>n, k<0, or n<0.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 5, 3, 1, 1, 3, 9, 8, 8, 3, 1, 1, 4, 12, 14, 16, 9, 4, 1, 1, 4, 16, 20, 30, 19, 13, 4, 1, 1, 5, 20, 30, 50, 39, 32, 14, 5, 1, 1, 5, 25, 40, 80, 69, 71, 36, 19, 5, 1, 1, 6, 30, 55, 120, 119, 140, 85, 55, 20, 6, 1

Offset: 0

Author

Michel Marcus, Sep 14 2016

Keywords

• nonn
• tabl

Comments

This is the triangle frst(n,k) in the Ehrenborg and Readdy link. See Definition 3.1 and Table 1.

Examples

			Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 2, 1;
  1, 2, 4, 2, 1;
  1, 3, 6, 5, 3, 1;
  1, 3, 9, 8, 8, 3, 1;
  ...
		

Links

• Richard Ehrenborg and Margaret A. Readdy, The Gaussian coefficient revisited, arXiv:1609.03216 [math.CO], 2016.

Crossrefs

Cf. A169623 (the triangle er).

Programs

Mathematica

T[n_, n_] = T[, 0] = 1; T[n, k_] /; 0 <= k <= n := T[n, k] = If[OddQ[k], T[n-1, k-1] + T[n-2, k], T[n-1, k-1] + T[n-1, k]]; T[, ] = 0;
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 28 2018 *)

PARI

frst(n, k) = if ((k>n) || (n<0) || (k<0), 0, if (n<=2, 1, if (k==0, 1, if (k%2, frst(n-1, k-1) + frst(n-2, k), frst(n-1, k-1) + frst(n-1, k)))));
tf(nn) = for (n=0, nn, for (k=0, n, print1(frst(n,k), ", ");); print(););