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A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A209599 #16 Dec 20 2017 03:19:19
%S A209599 1,2,0,3,1,0,5,3,0,0,8,7,1,0,0,13,15,4,0,0,0,21,30,12,1,0,0,0,34,58,
%T A209599 31,5,0,0,0,0,55,109,73,18,1,0,0,0,0,89,201,162,54,6,0,0,0,0,0,144,
%U A209599 365,344,145,25,1,0,0,0,0,0
%N A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A209599 A skew version of A122075.
%H A209599 G. C. Greubel, <a href="/A209599/b209599.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209599 G.f.: (1+x)/(1-x-(1+y)*x^2).
%F A209599 T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
%F A209599 Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
%e A209599 Triangle begins :
%e A209599   1
%e A209599   2, 0
%e A209599   3, 1, 0
%e A209599   5, 3, 0, 0
%e A209599   8, 7, 1, 0, 0
%e A209599   13, 15, 4, 0, 0, 0
%e A209599   21, 30, 12, 1, 0, 0, 0
%e A209599   34, 58, 31, 5, 0, 0, 0, 0
%e A209599   55, 109, 73, 18, 1, 0, 0, 0, 0
%e A209599   89, 201, 162, 54, 6, 0, 0, 0, 0, 0
%e A209599   144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
%e A209599   ...
%t A209599 T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Dec 19 2017 *)
%Y A209599 Cf. A122075, A122950, A000045, A023610, A129707
%K A209599 easy,nonn,tabl
%O A209599 0,2
%A A209599 _Philippe Deléham_, Mar 10 2012