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

1,5

Comments

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A088305, A186314.

Sequence in context: A169660 A035348 A140945 * A166960 A155908 A105373

Adjacent sequences: A141685 A141686 A141687 * A141689 A141690 A141691

Keyword

nonn,tabl

Author

Roger L. Bagula, Sep 09 2008

Extensions

Edited by G. C. Greubel, Mar 29 2021

Status

approved