A173882 Triangle T(n, k) = A090443(n-1)/(A090443(k-1)*A090443(n-k-1)) read by rows.

1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 60, 240, 60, 1, 1, 120, 1200, 1200, 120, 1, 1, 210, 4200, 10500, 4200, 210, 1, 1, 336, 11760, 58800, 58800, 11760, 336, 1, 1, 504, 28224, 246960, 493920, 246960, 28224, 504, 1, 1, 720, 60480, 846720, 2963520, 2963520, 846720, 60480, 720, 1

Offset: 0

Author

Roger L. Bagula, Mar 01 2010

Keywords

• nonn
• tabl
• easy

Comments

A090443 is defined as +1 at negative indices here, which keeps the definition valid in the range 0 <= k <= n.

Row sums are 1, 2, 8, 50, 362, 2642, 19322, 141794, 1045298, 7742882, ....

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   6,      1;
  1,  24,     24,       1;
  1,  60,    240,      60,        1;
  1, 120,   1200,    1200,      120,        1;
  1, 210,   4200,   10500,     4200,      210,        1;
  1, 336,  11760,   58800,    58800,    11760,      336,       1;
  1, 504,  28224,  246960,   493920,   246960,    28224,     504,      1;
  1, 720,  60480,  846720,  2963520,  2963520,   846720,   60480,    720,   1;
  1, 990, 118800, 2494800, 13970880, 24449040, 13970880, 2494800, 118800, 990, 1;
  ...
		

Links

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

Crossrefs

Cf. A056939.

Programs

Magma

T:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n-1,k-1)*Binomial(n,k)*Binomial(n+1,k+1)*(n-k)/(n-k+1) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 17 2021

Maple

A090443 := proc(n) (n+2)!*(n+1)!*n!/2 ; end proc:
A173882 := proc(n,m) if m=0 or m= n then 1; else A090443(n-1)/A090443(m-1)/A090443(n-m-1) ; end if; end proc:
seq(seq(A173882(n,m),m=0..n),n=0..5) ; # R. J. Mathar, Mar 19 2011

Mathematica

T[n_,k_]:= If[k==0||k==n, 1, 2*Binomial[n-1,k-1]*Binomial[n,k]*Binomial[n+1,k+1]*(n-k)/(n-k+1)];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 17 2021 *)

Sage

def T(n,k): return 1 if (k==0 or k==n) else 2*binomial(n-1,k-1)*binomial(n,k)*binomial(n+1,k+1)*(n-k)/(n-k+1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 17 2021

Formula

T(n, k) = 2*binomial(n-1,k-1)*binomial(n,k)*binomial(n+1,k+1)*(n-k)/(n-k+1) with T(n, 0) = T(n, n) = 1.

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