OFFSET
0,3
LINKS
G. C. Greubel, Antidiagonal rows n = 0..25, flattened
FORMULA
T(n, k) = q-binomial(2*n, n, k+1), where q-binomial(n, k, q) = Product_{j=0..k-1} ( (1-q^(n-j))/(1-q^(j+1)) ), read by antidiagonals. - G. C. Greubel, Jun 14 2021
EXAMPLE
Square array begins as:
1, 1, 1, 1, ...;
2, 3, 4, 5, ...;
6, 35, 130, 357, ...;
20, 1395, 33880, 376805, ...;
70, 200787, 75913222, 6221613541, ...;
252, 109221651, 1506472167928, 1634141006295525, ...;
Antidiagonal triangle begins as:
1;
1, 2;
1, 3, 6;
1, 4, 35, 20;
1, 5, 130, 1395, 70;
1, 6, 357, 33880, 200787, 252;
1, 7, 806, 376805, 75913222, 109221651, 924;
1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432;
MATHEMATICA
T[n_, k_]:= QBinomial[2*n, n, k+1];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)
PROG
(Magma)
QBinomial:= func< n, k, q | q eq 1 select Binomial(n, k) else k eq 0 select 1 else (&*[ (1-q^(n-j+1))/(1-q^j): j in [1..k] ]) >;
T:= func< n, k | QBinomial(2*n, n, k+1) >;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2021
(Sage)
def A156914(n, k): return q_binomial(2*n, n, k+1)
flatten([[A156914(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 18 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 14 2021
STATUS
approved