A173890 Triangle 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, read by rows.

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

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

Cf. A173889.

Sequence in context: A058293 A172092 A156888 * A159767 A169658 A330199

Adjacent sequences: A173887 A173888 A173889 * A173891 A173892 A173893

Keyword

nonn,tabl,less,easy

Author

Roger L. Bagula, Mar 01 2010

Extensions

Edited by G. C. Greubel, Apr 16 2021

Status

approved