A174451 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 = 3, read by rows.

1, 1, 1, 1, 18, 1, 1, 36, 36, 1, 1, 60, 2160, 60, 1, 1, 90, 5400, 5400, 90, 1, 1, 126, 11340, 680400, 11340, 126, 1, 1, 168, 21168, 1905120, 1905120, 21168, 168, 1, 1, 216, 36288, 4572288, 411505920, 4572288, 36288, 216, 1, 1, 270, 58320, 9797760, 1234517760, 1234517760, 9797760, 58320, 270, 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 = 3.

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 = 3.

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 = 3. (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,  18,     1;
  1,  36,    36,       1;
  1,  60,  2160,      60,          1;
  1,  90,  5400,    5400,         90,          1;
  1, 126, 11340,  680400,      11340,        126,       1;
  1, 168, 21168, 1905120,    1905120,      21168,     168,     1;
  1, 216, 36288, 4572288,  411505920,    4572288,   36288,   216,   1;
  1, 270, 58320, 9797760, 1234517760, 1234517760, 9797760, 58320, 270, 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, 3], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
F:= Factorial; // T = A174451
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, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 29 2021
(Sage)
f=factorial
def A174451(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([[A174451(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 29 2021

Crossrefs

Cf. A174449 (q=1), A174450 (q=2), this sequence (q=3).

Cf. A000108.

Sequence in context: A040324 A168623 A146774 * A144405 A202671 A203004

Adjacent sequences: A174448 A174449 A174450 * A174452 A174453 A174454

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 20 2010

Extensions

Edited by G. C. Greubel, Nov 29 2021

Status

approved