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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
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
KEYWORD
nonn,tabl,less,easy
AUTHOR
Roger L. Bagula, Mar 01 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 16 2021
STATUS
approved

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Last modified February 22 11:36 EST 2024. Contains 370255 sequences. (Running on oeis4.)