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A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0 with T(n,n) = 1 for n>=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.

%I A141751 #9 Jun 14 2017 01:08:05
%S A141751 1,1,1,2,2,1,5,5,3,1,13,13,8,4,1,34,34,21,11,5,1,89,89,55,29,14,6,1,
%T A141751 233,233,144,76,37,17,7,1,610,610,377,199,97,45,20,8,1,1597,1597,987,
%U A141751 521,254,118,53,23,9,1,4181,4181,2584,1364,665,309,139,61,26,10,1
%N A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0<k<n with T(n,n) = 1 for n>=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.
%H A141751 Paul D. Hanna, <a href="/A141751/b141751.txt">Table of n, a(n) for n = 0..1080, as a flattened table of rows 0..45</a>
%F A141751 T(n,k) = Fibonacci(2*(n-k)-1) + k*Fibonacci(2*(n-k)) for 0<=k<=n.
%e A141751 Generating rule.
%e A141751 Given nonzero elements W, X, Y, Z, relatively arranged like so:
%e A141751 .. W .....
%e A141751 .. X Y ...
%e A141751 .... Z ...
%e A141751 then Z = (X*Y + 1)/W.
%e A141751 Triangle begins:
%e A141751 1;
%e A141751 1, 1;
%e A141751 2, 2, 1;
%e A141751 5, 5, 3, 1;
%e A141751 13, 13, 8, 4, 1;
%e A141751 34, 34, 21, 11, 5, 1;
%e A141751 89, 89, 55, 29, 14, 6, 1;
%e A141751 233, 233, 144, 76, 37, 17, 7, 1;
%e A141751 610, 610, 377, 199, 97, 45, 20, 8, 1;
%e A141751 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
%e A141751 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...
%o A141751 (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(k==0,fibonacci(2*n-1),(T(n-1,k-1)*T(n-1,k)+1)/T(n-2,k-1))))
%o A141751 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%o A141751 (PARI) T(n,k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
%o A141751 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A141751 Cf. row sums: A141752; columns: A001519, A001906, A002878, A054486, A054492, A055267, A055273, A055849.
%K A141751 nonn,tabl
%O A141751 0,4
%A A141751 _Paul D. Hanna_, Jul 04 2008