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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.
1

%I #6 Mar 30 2021 01:51:02

%S 1,1,1,1,6,1,1,26,26,1,1,99,416,99,1,1,352,5407,5407,352,1,1,1200,

%T 62616,227094,62616,1200,1,1,3977,673728,8212854,8212854,673728,3977,

%U 1,1,12918,6889153,269486766,903413940,269486766,6889153,12918,1,1,41338,67863290,8256432767,88493861004,88493861004,8256432767,67863290,41338,1

%N 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 Row sums are: {1, 2, 8, 54, 616, 11520, 354728, 17781120, 1456191616, 193636396800, ...}.

%H G. C. Greubel, <a href="/A141688/b141688.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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 From _G. C. Greubel_, Mar 29 2021: (Start)

%F 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 T(n, 2) = A186314(n+1). (End)

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 6, 1;

%e 1, 26, 26, 1;

%e 1, 99, 416, 99, 1;

%e 1, 352, 5407, 5407, 352, 1;

%e 1, 1200, 62616, 227094, 62616, 1200, 1;

%e 1, 3977, 673728, 8212854, 8212854, 673728, 3977, 1;

%e 1, 12918, 6889153, 269486766, 903413940,269486766, 6889153, 12918, 1;

%t (* First program *)

%t b[n_]:= b[n]= If[n==0, 1, Sum[k*b[n-k], {k,n}]];

%t 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 Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 29 2021 *)

%t (* Second program *)

%t 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 Table[T[n, k], {n, 12}, {k, n}]//Flatten (* _G. C. Greubel_, Mar 29 2021 *)

%o (Magma)

%o function T(n,k)

%o if k eq 1 or k eq n then return 1;

%o else return Fibonacci(2*(n-k+1))*T(n-1, k-1) + Fibonacci(2*k)*T(n-1, k);

%o end if; return T;

%o end function;

%o [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 29 2021

%o (Sage)

%o @CachedFunction

%o 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 flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 29 2021

%Y Cf. A088305, A186314.

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_, Sep 09 2008

%E Edited by _G. C. Greubel_, Mar 29 2021