A174450 Triangle T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 2, read by rows.

1, 1, 1, 1, 12, 1, 1, 24, 24, 1, 1, 40, 960, 40, 1, 1, 60, 2400, 2400, 60, 1, 1, 84, 5040, 201600, 5040, 84, 1, 1, 112, 9408, 564480, 564480, 9408, 112, 1, 1, 144, 16128, 1354752, 81285120, 1354752, 16128, 144, 1, 1, 180, 25920, 2903040, 243855360, 243855360, 2903040, 25920, 180, 1

Offset

0,5

Links

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

Formula

T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 2.

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

From G. C. Greubel, Nov 29 2021: (Start)

T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 2.

T(n, k, q) = binomial(n, k)*binomial(n+1, k+1) * ( k!*(k+1)!*q^k/(n-k+1) if (floor(n/2) >= k), otherwise ((n-k)!)^2*q^(n-k) ), for q = 2. (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,  12,     1;
  1,  24,    24,       1;
  1,  40,   960,      40,         1;
  1,  60,  2400,    2400,        60,         1;
  1,  84,  5040,  201600,      5040,        84,       1;
  1, 112,  9408,  564480,    564480,      9408,     112,     1;
  1, 144, 16128, 1354752,  81285120,   1354752,   16128,   144,   1;
  1, 180, 25920, 2903040, 243855360, 243855360, 2903040, 25920, 180,  1;

Mathematica

T[n_, k_, q_]:= If[Floor[n/2]>k-1, n!*(n+1)!*q^k/((n-k)!*(n-k+1)!), n!*(n+1)!*q^(n-k)/(k!*(k+1)!)];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
F:= Factorial; // T = A174450
T:= func< n, k, q | Floor(n/2) gt k-1 select F(n)*F(n+1)*q^k/(F(n-k)*F(n-k+1)) else F(n)*F(n+1)*q^(n-k)/(F(k)*F(k+1)) >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 29 2021
(Sage)
f=factorial
def A174450(n, k, q):
    if ((n//2)>k-1): return f(n)*f(n+1)*q^k/(f(n-k)*f(n-k+1))
    else: return f(n)*f(n+1)*q^(n-k)/(f(k)*f(k+1))
flatten([[A174450(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 29 2021

Crossrefs

Cf. A174449 (q=1), this sequence (q=2), A174451 (q=3).

Cf. A000108.

Sequence in context: A070636 A168646 A051457 * A166343 A186432 A176489

Adjacent sequences: A174447 A174448 A174449 * A174451 A174452 A174453

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 20 2010

Extensions

Edited by G. C. Greubel, Nov 29 2021

Status

approved