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

%I A156914 #12 Sep 08 2022 08:45:41
%S A156914 1,1,2,1,3,6,1,4,35,20,1,5,130,1395,70,1,6,357,33880,200787,252,1,7,
%T A156914 806,376805,75913222,109221651,924,1,8,1591,2558556,6221613541,
%U A156914 1506472167928,230674393235,3432,1,9,2850,12485095,200525284806,1634141006295525,267598665689058580,1919209135381395,12870
%N A156914 Square array T(n, k) = q-binomial(2*n, n, k+1), read by antidiagonals.
%H A156914 G. C. Greubel, <a href="/A156914/b156914.txt">Antidiagonal rows n = 0..25, flattened</a>
%F A156914 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 A156914 Square array begins as:
%e A156914     1,         1,             1,                1, ...;
%e A156914     2,         3,             4,                5, ...;
%e A156914     6,        35,           130,              357, ...;
%e A156914    20,      1395,         33880,           376805, ...;
%e A156914    70,    200787,      75913222,       6221613541, ...;
%e A156914   252, 109221651, 1506472167928, 1634141006295525, ...;
%e A156914 Antidiagonal triangle begins as:
%e A156914   1;
%e A156914   1, 2;
%e A156914   1, 3,    6;
%e A156914   1, 4,   35,      20;
%e A156914   1, 5,  130,    1395,         70;
%e A156914   1, 6,  357,   33880,     200787,           252;
%e A156914   1, 7,  806,  376805,   75913222,     109221651,          924;
%e A156914   1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432;
%t A156914 T[n_, k_]:= QBinomial[2*n, n, k+1];
%t A156914 Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 14 2021 *)
%o A156914 (Magma)
%o A156914 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 A156914 T:= func< n,k | QBinomial(2*n, n, k+1) >;
%o A156914 [T(k, n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 14 2021
%o A156914 (Sage)
%o A156914 def A156914(n, k): return q_binomial(2*n, n, k+1)
%o A156914 flatten([[A156914(k,n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 14 2021
%Y A156914 Cf. A000984, A022166, A022167, A022168, A022169, A022170, A022171, A022175.
%K A156914 nonn,tabl
%O A156914 0,3
%A A156914 _Roger L. Bagula_, Feb 18 2009
%E A156914 Edited by _G. C. Greubel_, Jun 14 2021