A360208 Triangular array T(n,k) read by antidiagonals T(n,k) = F(n)!/(F(k)!*F(n-k)!), where F(m) = A000045(m) = m-th Fibonacci number.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 20, 60, 60, 20, 1, 1, 336, 6720, 10080, 6720, 336, 1, 1, 154440, 51891840, 518918400, 518918400, 51891840, 154440, 1, 1, 8204716800, 1267136462592000, 212878925715456000, 1419192838103040000, 212878925715456000, 1267136462592000, 8204716800, 1

Offset

0,8

Comments

Analogous to Pascal's triangle, A007318.

Links

G. C. Greubel, Rows n = 0..14 of the triangle, flattened

Formula

T(n,k) = F(n)!/(F(k)!*F(n - k)!), where F(m) = A000045(m) = m-th Fibonacci number.

T(n, n-k) = T(n, k). - G. C. Greubel, Jun 21 2024

Example

First seven rows:
  1
  1    1
  1    1     1
  1    2     2      1
  1    3     6      3     1
  1   20    60     60    20    1
  1  336  6720  10080  6720  336   1
  ...

Maple

F:= combinat[fibonacci]:
T:= (n, k)-> F(n)!/(F(k)!*F(n-k)!):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Jan 30 2023

Mathematica

f[n_]:= Fibonacci[n]!;
t = Table[f[n]/(f[k]*f[n-k]), {n, 0, 8}, {k, 0, n}];
TableForm[t]   (* array *)
Flatten[t]     (* sequence *)

Prog

(Magma)
F:= func< n | Factorial(Fibonacci(n)) >;
[F(n)/(F(k)*F(n-k)): k in [0..n], n in [0..10]]; // G. C. Greubel, Jun 21 2024
(SageMath)
def f(n): return factorial(fibonacci(n))
flatten([[f(n)/(f(k)*f(n-k)) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Jun 21 2024

Crossrefs

Cf. A000045, A007318, A360207.

Sequence in context: A156133 A010048 A055870 * A360571 A088459 A300699

Adjacent sequences: A360205 A360206 A360207 * A360209 A360210 A360211

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 30 2023

Status

approved