A300945 Rectangular array A(n, k) = hypergeom([-k, k + n/2 - 1], [1], -4) with row n >= 0 and k >= 0, read by ascending antidiagonals.
1, 1, 1, 1, 3, 25, 1, 5, 43, 425, 1, 7, 65, 661, 7025, 1, 9, 91, 965, 10515, 116625, 1, 11, 121, 1345, 15105, 171097, 1951625, 1, 13, 155, 1809, 20995, 243525, 2828101, 32903225, 1, 15, 193, 2365, 28401, 337877, 4001345, 47284251, 558265825
Offset: 0
Examples
[0] 1, 1, 25, 425, 7025, 116625, 1951625, 32903225, ... [A299845] [1] 1, 3, 43, 661, 10515, 171097, 2828101, 47284251, ... [A299506] [2] 1, 5, 65, 965, 15105, 243525, 4001345, 66622085, ... [3] 1, 7, 91, 1345, 20995, 337877, 5544709, 92234527, ... [A243946] [4] 1, 9, 121, 1809, 28401, 458649, 7544041, 125700129, ... [A084769] [5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947] [6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...
Programs
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Mathematica
Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}]; Table[Print[Arow[n, 7]], {n, 0, 6}]; 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 *)
Formula
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