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A141688 Triangle T(n, k) = Fibonacci(2*k)*T(n-1, k) + Fibonacci(2*(n-k+1))*T(n-1, k-1), with T(n, 1) = T(n, n) = 1, read by rows.

%I A141688 #6 Mar 30 2021 01:51:02
%S A141688 1,1,1,1,6,1,1,26,26,1,1,99,416,99,1,1,352,5407,5407,352,1,1,1200,
%T A141688 62616,227094,62616,1200,1,1,3977,673728,8212854,8212854,673728,3977,
%U A141688 1,1,12918,6889153,269486766,903413940,269486766,6889153,12918,1,1,41338,67863290,8256432767,88493861004,88493861004,8256432767,67863290,41338,1
%N A141688 Triangle T(n, k) = Fibonacci(2*k)*T(n-1, k) + Fibonacci(2*(n-k+1))*T(n-1, k-1), with T(n, 1) = T(n, n) = 1, read by rows.
%C A141688 Row sums are: {1, 2, 8, 54, 616, 11520, 354728, 17781120, 1456191616, 193636396800, ...}.
%H A141688 G. C. Greubel, <a href="/A141688/b141688.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A141688 Let A088305(n) be defined by b(n) = Sum_{j=1..n} j*b(n-j), with b(0)=1, then T(n, k) = b(n-k+1)*T(n-1, k-1) + b(k)*T(n-1, k) with T(n,1) = T(n,n) = 1.
%F A141688 From _G. C. Greubel_, Mar 29 2021: (Start)
%F A141688 T(n, k) = Fibonacci(2*k)*T(n-1, k) + Fibonacci(2*(n-k+1))*T(n-1, k-1), with T(n, 1) = T(n, n) = 1.
%F A141688 T(n, 2) = A186314(n+1). (End)
%e A141688 Triangle begins as:
%e A141688   1;
%e A141688   1,     1;
%e A141688   1,     6,       1;
%e A141688   1,    26,      26,         1;
%e A141688   1,    99,     416,        99,         1;
%e A141688   1,   352,    5407,      5407,       352,        1;
%e A141688   1,  1200,   62616,    227094,     62616,     1200,       1;
%e A141688   1,  3977,  673728,   8212854,   8212854,   673728,    3977,     1;
%e A141688   1, 12918, 6889153, 269486766, 903413940,269486766, 6889153, 12918, 1;
%t A141688 (* First program *)
%t A141688 b[n_]:= b[n]= If[n==0, 1, Sum[k*b[n-k], {k,n}]];
%t A141688 T[n_, k_]:= If[k==1 || k==n, 1, b[n-k+1]*T[n-1, k-1] + b[k]*T[n-1, k]];
%t A141688 Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 29 2021 *)
%t A141688 (* Second program *)
%t A141688 T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, Fibonacci[2*(n-k+1)]*T[n-1, k-1] + Fibonacci[2*k]*T[n-1, k]];
%t A141688 Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Mar 29 2021 *)
%o A141688 (Magma)
%o A141688 function T(n,k)
%o A141688   if k eq 1 or k eq n then return 1;
%o A141688   else return Fibonacci(2*(n-k+1))*T(n-1, k-1) + Fibonacci(2*k)*T(n-1, k);
%o A141688   end if; return T;
%o A141688 end function;
%o A141688 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 29 2021
%o A141688 (Sage)
%o A141688 @CachedFunction
%o A141688 def T(n,k): return 1 if (k==1 or k==n) else fibonacci(2*(n-k+1))*T(n-1, k-1) + fibonacci(2*k)*T(n-1, k)
%o A141688 flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 29 2021
%Y A141688 Cf. A088305, A186314.
%K A141688 nonn,tabl
%O A141688 1,5
%A A141688 _Roger L. Bagula_, Sep 09 2008
%E A141688 Edited by _G. C. Greubel_, Mar 29 2021