A156914 Square array T(n, k) = q-binomial(2*n, n, k+1), read by antidiagonals.

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, 1, 9, 2850, 12485095, 200525284806, 1634141006295525, 267598665689058580, 1919209135381395, 12870

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

Cf. A000984, A022166, A022167, A022168, A022169, A022170, A022171, A022175.

Sequence in context: A214306 A337411 A337407 * A289656 A248686 A059434

Adjacent sequences: A156911 A156912 A156913 * A156915 A156916 A156917

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 18 2009

Extensions

Edited by G. C. Greubel, Jun 14 2021

Status

approved