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 = 1.
T(n, n-k, q) = T(n, k, q).
From G. C. Greubel, Nov 29 2021: (Start)
T(2*n, n, 1) = A052510(n+1).
T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 1.
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 = 1. (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 12, 12, 1;
1, 20, 240, 20, 1;
1, 30, 600, 600, 30, 1;
1, 42, 1260, 25200, 1260, 42, 1;
1, 56, 2352, 70560, 70560, 2352, 56, 1;
1, 72, 4032, 169344, 5080320, 169344, 4032, 72, 1;
1, 90, 6480, 362880, 15240960, 15240960, 362880, 6480, 90, 1;
1, 110, 9900, 712800, 39916800, 1676505600, 39916800, 712800, 9900, 110, 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, 1], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
F:= Factorial; // T = A174449
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, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 29 2021
(Sage)
f=factorial
def A174449(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([[A174449(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 29 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 20 2010
EXTENSIONS
Edited by G. C. Greubel, Nov 29 2021
STATUS
approved