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Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.
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%I #13 May 30 2024 00:29:13

%S 1,1,1,1,3,25,1,5,43,425,1,7,65,661,7025,1,9,91,965,10515,116625,1,11,

%T 121,1345,15105,171097,1951625,1,13,155,1809,20995,243525,2828101,

%U 32903225,1,15,193,2365,28401,337877,4001345,47284251,558265825

%N Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.

%F T(n, k) = if k = 0 then 1, otherwise 4^k*Sum_{j=0..n} (5/4)^j * binomial(k, j) * binomial(k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)). - _Detlef Meya_, May 28 2024

%e [0] 1, 1, 25, 425, 7025, 116625, 1951625, 32903225, ... [A299845]

%e [1] 1, 3, 43, 661, 10515, 171097, 2828101, 47284251, ... [A299506]

%e [2] 1, 5, 65, 965, 15105, 243525, 4001345, 66622085, ...

%e [3] 1, 7, 91, 1345, 20995, 337877, 5544709, 92234527, ... [A243946]

%e [4] 1, 9, 121, 1809, 28401, 458649, 7544041, 125700129, ... [A084769]

%e [5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947]

%e [6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...

%t Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}];

%t Table[Print[Arow[n, 7]], {n, 0, 6}];

%t T[n_, k_] := If[k==0, 1, 4^k*Sum[(5/4)^j*Binomial[k, j]*Binomial[k - 2 + ((n - k)/2), j - 2 + ((n - k)/2)] ,{j, 0, n}]]; Flatten[Table[T[n, k],{n, 0, 8}, {k, 0, n}]] (* _Detlef Meya_, May 28 2024 *)

%Y Cf. A299845, A299506, A243946, A084769, A243947.

%Y Cf. A300946.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Mar 16 2018