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)!*(n+3)!/1440 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)*(j+3) with c(0) = c(1) = 1 and c(2) = 2.
EXAMPLE
The triangle begins as:
1;
1, 1;
1, 2, 1;
1, 72, 72, 1;
1, 840, 30240, 840, 1;
1, 2880, 1209600, 1209600, 2880, 1;
1, 7560, 10886400, 127008000, 10886400, 7560, 1;
1, 16800, 63504000, 2540160000, 2540160000, 63504000, 16800, 1;
1, 33264, 279417600, 29338848000, 100590336000, 29338848000, 279417600, 33264, 1;
MATHEMATICA
c[n_]:= c[n]= If[n<3, Fibonacci[n+1], (n-2)!*(n-1)!*n!*(n+1)!*(n+3)!/1440 ];
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 le 2 select Fibonacci(n+1) else F(n-2)*F(n-1)*F(n)*F(n+1)*F(n+3)/1440 >;
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)*f(n+3)/1440
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
KEYWORD
AUTHOR
Roger L. Bagula, Mar 01 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 16 2021
STATUS
approved