A173889 - OEIS

%I #8 Apr 17 2021 02:11:42

%S 1,1,1,1,2,1,1,12,12,1,1,120,720,120,1,1,360,21600,21600,360,1,1,840,

%T 151200,1512000,151200,840,1,1,1680,705600,21168000,21168000,705600,

%U 1680,1,1,3024,2540160,177811200,533433600,177811200,2540160,3024,1

%N 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.

%H G. C. Greubel, <a href="/A173889/b173889.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F 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.

%e The triangle begins as:

%e   1;

%e   1,    1;

%e   1,    2,       1;

%e   1,   12,      12,         1;

%e   1,  120,     720,       120,         1;

%e   1,  360,   21600,     21600,       360,         1;

%e   1,  840,  151200,   1512000,    151200,       840,       1;

%e   1, 1680,  705600,  21168000,  21168000,    705600,    1680,    1;

%e   1, 3024, 2540160, 177811200, 533433600, 177811200, 2540160, 3024, 1;

%t c[n_]:= c[n]= If[n<3, Fibonacci[n+1], (n-2)!*(n-1)!*n!*(n+1)!/12 ];

%t T[n_, k_]:= c[n]/(c[k]*c[n-k]);

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 16 2021 *)

%o (Magma)

%o F:=Factorial;

%o c:= func< n | n eq 2 select Fibonacci(n+1) else F(n-2)*F(n-1)*F(n)*F(n+1)/12 >;

%o T:= func< n,k | c(n)/(c(k)*c(n-k)) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 16 2021

%o (Sage)

%o f=factorial

%o @CachedFunction

%o def c(n): return fibonacci(n+1) if (n<3) else f(n-2)*f(n-1)*f(n)*f(n+1)/12

%o def T(n, k): return c(n)/(c(k)*c(n-k))

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 16 2021

%Y Cf. A173890.

%K nonn,tabl,less,easy

%O 0,5

%A _Roger L. Bagula_, Mar 01 2010

%E Edited by _G. C. Greubel_, Apr 16 2021