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

1,5

Links

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

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).

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A081582.

Sequence in context: A168625 A143681 A081582 * A144404 A014761 A073702

Adjacent sequences: A174343 A174344 A174345 * A174347 A174348 A174349

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 16 2010

Extensions

Edited by G. C. Greubel, Nov 26 2021

Status

approved