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.

1, 1, 1, 1, 6, 1, 1, 26, 26, 1, 1, 99, 416, 99, 1, 1, 352, 5407, 5407, 352, 1, 1, 1200, 62616, 227094, 62616, 1200, 1, 1, 3977, 673728, 8212854, 8212854, 673728, 3977, 1, 1, 12918, 6889153, 269486766, 903413940, 269486766, 6889153, 12918, 1, 1, 41338, 67863290, 8256432767, 88493861004, 88493861004, 8256432767, 67863290, 41338, 1

Offset: 1

Author

Roger L. Bagula, Sep 09 2008

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins as:
  1;
  1,     1;
  1,     6,       1;
  1,    26,      26,         1;
  1,    99,     416,        99,         1;
  1,   352,    5407,      5407,       352,        1;
  1,  1200,   62616,    227094,     62616,     1200,       1;
  1,  3977,  673728,   8212854,   8212854,   673728,    3977,     1;
  1, 12918, 6889153, 269486766, 903413940,269486766, 6889153, 12918, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A088305, A186314.

Programs

Magma

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return Fibonacci(2*(n-k+1))*T(n-1, k-1) + Fibonacci(2*k)*T(n-1, k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 29 2021

Mathematica

(* First program *)
b[n_]:= b[n]= If[n==0, 1, Sum[k*b[n-k], {k,n}]];
T[n_, k_]:= If[k==1 || k==n, 1, b[n-k+1]*T[n-1, k-1] + b[k]*T[n-1, k]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 29 2021 *)
(* Second program *)
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]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 29 2021 *)

Sage

@CachedFunction
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)
flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 29 2021

Formula

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.

From G. C. Greubel, Mar 29 2021: (Start)

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.

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

Extensions

Edited by G. C. Greubel, Mar 29 2021