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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.

%I A276696 #12 Jun 11 2022 11:41:40
%S A276696 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,
%T A276696 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,
%U A276696 69,71,36,19,5,1,1,6,30,55,120,119,140,85,55,20,6,1
%N 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.
%C A276696 This is the triangle frst(n,k) in the Ehrenborg and Readdy link. See Definition 3.1 and Table 1.
%H A276696 Richard Ehrenborg and Margaret A. Readdy, <a href="http://arxiv.org/abs/1609.03216">The Gaussian coefficient revisited</a>, arXiv:1609.03216 [math.CO], 2016.
%e A276696 Triangle starts:
%e A276696   1;
%e A276696   1, 1;
%e A276696   1, 1, 1;
%e A276696   1, 2, 2, 1;
%e A276696   1, 2, 4, 2, 1;
%e A276696   1, 3, 6, 5, 3, 1;
%e A276696   1, 3, 9, 8, 8, 3, 1;
%e A276696   ...
%t A276696 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;
%t A276696 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 28 2018 *)
%o A276696 (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)))));
%o A276696 tf(nn) = for (n=0, nn, for (k=0, n, print1(frst(n,k), ", ");); print(););
%Y A276696 Cf. A169623 (the triangle er).
%K A276696 nonn,tabl
%O A276696 0,8
%A A276696 _Michel Marcus_, Sep 14 2016