A174346 Triangle T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ), read by rows.

1, 1, 1, 1, 9, 1, 1, 18, 18, 1, 1, 30, 180, 30, 1, 1, 45, 450, 450, 45, 1, 1, 63, 945, 4725, 945, 63, 1, 1, 84, 1764, 13230, 13230, 1764, 84, 1, 1, 108, 3024, 31752, 142884, 31752, 3024, 108, 1, 1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1

Offset: 1

Author

Roger L. Bagula, Mar 16 2010

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   9,    1;
  1,  18,   18,     1;
  1,  30,  180,    30,      1;
  1,  45,  450,   450,     45,      1;
  1,  63,  945,  4725,    945,     63,     1;
  1,  84, 1764, 13230,  13230,   1764,    84,    1;
  1, 108, 3024, 31752, 142884,  31752,  3024,  108,   1;
  1, 135, 4860, 68040, 428652, 428652, 68040, 4860, 135, 1;
		

Links

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

Crossrefs

Cf. A081582.

Programs

Magma

function T(n,k)
  if Floor(n/2) gt k-1 then return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(k-1);
  else return (1/n)*Binomial(n,k)*Binomial(n,k-1)*3^(n-k);
  end if; return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 26 2021

Mathematica

T[n_,k_]:= (Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>k-1, 3^(k-1), 3^(n-k)];
Table[T[n,k], {n,12}, {k,n}]//Flatten

Sage

def A174346(n,k): return (1/n)*binomial(n,k)*binomial(n,k-1)*( 3^(k-1) if ((n//2)>k-1) else 3^(n-k) )
flatten([[A174346(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Nov 26 2021

Formula

T(n, k) = (binomial(n-1, k-1)*binomial(n, k-1)/k) * ( 3^(k-1) if floor(n/2) >= k, otherwise 3^(n-k) ).

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

Extensions

Edited by G. C. Greubel, Nov 26 2021