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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).

%I A123603 #18 Dec 12 2018 12:37:01
%S A123603 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,
%T A123603 21,33,69,81,81,69,33,21,34,59,133,167,199,167,133,59,34,55,105,249,
%U A123603 345,435,435,345,249,105,55,89,185,462,687,945,1005,945,687,462,185,89
%N 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).
%H A123603 G. C. Greubel, <a href="/A123603/b123603.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A123603 T(n,k) = T(n,n-k).
%F A123603 T(n,0) = Fibonacci(n+1) = A000045(n+1).
%F A123603 T(n+1,1) = A010049(n+1).
%F A123603 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.
%F A123603 G.f.: 1/(1 - x - x*y - x^2 + x^2*y - x^2*y^2).
%e A123603 Triangle begins:
%e A123603 1;
%e A123603 1, 1;
%e A123603 2, 1, 2;
%e A123603 3, 3, 3, 3;
%e A123603 5, 5, 9, 5, 5;
%e A123603 8, 10, 17, 17, 10, 8;
%e A123603 13, 18, 36, 35, 36, 18, 13;
%e A123603 21, 33, 69, 81, 81, 69, 33, 21;
%e A123603 34, 59, 133, 167, 199, 167, 133, 59, 34;
%e A123603 55, 105, 249, 345, 435, 435, 345, 249, 105, 55;
%e A123603 89, 185, 462, 687, 945, 1005, 945, 687, 462, 185, 89; ...
%t A123603 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 A123603 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 *)
%Y A123603 Cf. A000045, A000129, A322239 (central terms).
%K A123603 nonn,tabl
%O A123603 0,4
%A A123603 _Philippe Deléham_, Nov 14 2006, Mar 14 2014