A174345 Triangle T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k), read by rows.

1, 1, 1, 1, 6, 1, 1, 12, 12, 1, 1, 20, 80, 20, 1, 1, 30, 200, 200, 30, 1, 1, 42, 420, 1400, 420, 42, 1, 1, 56, 784, 3920, 3920, 784, 56, 1, 1, 72, 1344, 9408, 28224, 9408, 1344, 72, 1, 1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90, 1

Offset: 1

Author

Roger L. Bagula, Mar 16 2010

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  6,    1;
  1, 12,   12,     1;
  1, 20,   80,    20,     1;
  1, 30,  200,   200,    30,     1;
  1, 42,  420,  1400,   420,    42,     1;
  1, 56,  784,  3920,  3920,   784,    56,    1;
  1, 72, 1344,  9408, 28224,  9408,  1344,   72,  1;
  1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90,  1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000217, A180291.

Programs

Mathematica

Table[(Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>=k, 2^(k-1), 2^(n-k)], {n,12}, {k,n}]//Flatten

Sage

def A174345(n,k):
    b=binomial
    if ((n//2)>k-1): return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(k-1)
    else: return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(n-k)
flatten([[A174345(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Nov 28 2021

Formula

T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k).

T(n, n-k) = T(n, k).

From G. C. Greubel, Nov 28 2021: (Start)

T(n, n-1) = A180291(n), n > 1.

T(n, n-1) = 2*A000217(n-1), n > 2. (End)

Extensions

Edited by G. C. Greubel, Nov 28 2021