A094444 Triangular array T(n,k) = Fibonacci(n+4-k)*C(n,k), k=0..n, n>=0.

3, 5, 3, 8, 10, 3, 13, 24, 15, 3, 21, 52, 48, 20, 3, 34, 105, 130, 80, 25, 3, 55, 204, 315, 260, 120, 30, 3, 89, 385, 714, 735, 455, 168, 35, 3, 144, 712, 1540, 1904, 1470, 728, 224, 40, 3, 233, 1296, 3204, 4620, 4284, 2646, 1092, 288, 45, 3, 377, 2330, 6480, 10680, 11550, 8568, 4410, 1560, 360, 50, 3

Offset

0,1

Comments

Row sums are Fibonacci numbers.

Row sums with alternating signs are Fibonacci numbers or their negatives.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

From G. C. Greubel, Oct 30 2019: (Start)

T(n,k) = binomial(n,k)*Fibonacci(n-k+4).

Sum_{k=0..n} T(n,k) = Fibonacci(2*n+4).

Sum_{k=0..n} (-1)^(k+1) * T(n,k) = (-1)^n * Fibonacci(n-4). (End)

Example

First few rows:
   3;
   5,   3;
   8,  10,   3;
  13,  24,  15,  3;
  21,  52,  48, 20,  3;
  34, 105, 130, 80, 25, 3;

Maple

with(combinat); seq(seq(fibonacci(n-k+4)*binomial(n, k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

Mathematica

Table[Fibonacci[n-k+4]*Binomial[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

Prog

(PARI) T(n, k) = binomial(n, k)*fibonacci(n-k+4);
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 30 2019
(Magma) [Binomial(n, k)*Fibonacci(n-k+4): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
(Sage) [[binomial(n, k)*fibonacci(n-k+4) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*Fibonacci(n-k+4) ))); # G. C. Greubel, Oct 30 2019

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094443.

Sequence in context: A354528 A328915 A100338 * A231641 A099446 A198827

Adjacent sequences: A094441 A094442 A094443 * A094445 A094446 A094447

Keyword

nonn,tabl

Author

Clark Kimberling, May 03 2004

Status

approved