A156914 - OEIS

A156914

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

%I #12 Sep 08 2022 08:45:41

%S 1,1,2,1,3,6,1,4,35,20,1,5,130,1395,70,1,6,357,33880,200787,252,1,7,

%T 806,376805,75913222,109221651,924,1,8,1591,2558556,6221613541,

%U 1506472167928,230674393235,3432,1,9,2850,12485095,200525284806,1634141006295525,267598665689058580,1919209135381395,12870

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

%H G. C. Greubel, <a href="/A156914/b156914.txt">Antidiagonal rows n = 0..25, flattened</a>

%F 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

%e Square array begins as:

%e 1, 1, 1, 1, ...;

%e 2, 3, 4, 5, ...;

%e 6, 35, 130, 357, ...;

%e 20, 1395, 33880, 376805, ...;

%e 70, 200787, 75913222, 6221613541, ...;

%e 252, 109221651, 1506472167928, 1634141006295525, ...;

%e Antidiagonal triangle begins as:

%e 1;

%e 1, 2;

%e 1, 3, 6;

%e 1, 4, 35, 20;

%e 1, 5, 130, 1395, 70;

%e 1, 6, 357, 33880, 200787, 252;

%e 1, 7, 806, 376805, 75913222, 109221651, 924;

%e 1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432;

%t T[n_, k_]:= QBinomial[2*n, n, k+1];

%t Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 14 2021 *)

%o (Magma)

%o 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] ]) >;

%o T:= func< n,k | QBinomial(2*n, n, k+1) >;

%o [T(k, n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 14 2021

%o (Sage)

%o def A156914(n, k): return q_binomial(2*n, n, k+1)

%o flatten([[A156914(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021

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

%K nonn,tabl

%O 0,3

%A _Roger L. Bagula_, Feb 18 2009

%E Edited by _G. C. Greubel_, Jun 14 2021