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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 12 21:39 EDT 2021. Contains 344968 sequences. (Running on oeis4.)