A173889 Triangle T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = (n-2)!*(n-1)!*n!*(n+1)!/12 with c(0) = c(1) = 1 and c(2) = 2, read by rows.

1, 1, 1, 1, 2, 1, 1, 12, 12, 1, 1, 120, 720, 120, 1, 1, 360, 21600, 21600, 360, 1, 1, 840, 151200, 1512000, 151200, 840, 1, 1, 1680, 705600, 21168000, 21168000, 705600, 1680, 1, 1, 3024, 2540160, 177811200, 533433600, 177811200, 2540160, 3024, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = (n-2)!*(n-1)!*n!*(n+1)!/12 with c(0) = c(1) = 1 and c(2) = 2.

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=3..n} (j-2)*(j-1)*j*(j+1) with c(0) = c(1) = 1 and c(2) = 2.

Example

The triangle begins as:
  1;
  1,    1;
  1,    2,       1;
  1,   12,      12,         1;
  1,  120,     720,       120,         1;
  1,  360,   21600,     21600,       360,         1;
  1,  840,  151200,   1512000,    151200,       840,       1;
  1, 1680,  705600,  21168000,  21168000,    705600,    1680,    1;
  1, 3024, 2540160, 177811200, 533433600, 177811200, 2540160, 3024, 1;

Mathematica

c[n_]:= c[n]= If[n<3, Fibonacci[n+1], (n-2)!*(n-1)!*n!*(n+1)!/12 ];
T[n_, k_]:= c[n]/(c[k]*c[n-k]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 16 2021 *)

Prog

(Magma)
F:=Factorial;
c:= func< n | n eq 2 select Fibonacci(n+1) else F(n-2)*F(n-1)*F(n)*F(n+1)/12 >;
T:= func< n, k | c(n)/(c(k)*c(n-k)) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 16 2021
(Sage)
f=factorial
@CachedFunction
def c(n): return fibonacci(n+1) if (n<3) else f(n-2)*f(n-1)*f(n)*f(n+1)/12
def T(n, k): return c(n)/(c(k)*c(n-k))
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

Crossrefs

Cf. A173890.

Sequence in context: A297762 A010246 A186430 * A156885 A174718 A176291

Adjacent sequences: A173886 A173887 A173888 * A173890 A173891 A173892

Keyword

nonn,tabl,less,easy

Author

Roger L. Bagula, Mar 01 2010

Extensions

Edited by G. C. Greubel, Apr 16 2021

Status

approved