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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.

%I A300945 #13 May 30 2024 00:29:13
%S A300945 1,1,1,1,3,25,1,5,43,425,1,7,65,661,7025,1,9,91,965,10515,116625,1,11,
%T A300945 121,1345,15105,171097,1951625,1,13,155,1809,20995,243525,2828101,
%U A300945 32903225,1,15,193,2365,28401,337877,4001345,47284251,558265825
%N 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.
%F A300945 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 A300945 [0] 1,  1,  25,  425,  7025, 116625,  1951625,  32903225, ... [A299845]
%e A300945 [1] 1,  3,  43,  661, 10515, 171097,  2828101,  47284251, ... [A299506]
%e A300945 [2] 1,  5,  65,  965, 15105, 243525,  4001345,  66622085, ...
%e A300945 [3] 1,  7,  91, 1345, 20995, 337877,  5544709,  92234527, ... [A243946]
%e A300945 [4] 1,  9, 121, 1809, 28401, 458649,  7544041, 125700129, ... [A084769]
%e A300945 [5] 1, 11, 155, 2365, 37555, 610897, 10098997, 168894355, ... [A243947]
%e A300945 [6] 1, 13, 193, 3021, 48705, 800269, 13324417, 224028877, ...
%t A300945 Arow[n_, len_] := Table[Hypergeometric2F1[-k, k + n/2 - 1, 1, -4], {k, 0, len}];
%t A300945 Table[Print[Arow[n, 7]], {n, 0, 6}];
%t A300945 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 A300945 Cf. A299845, A299506, A243946, A084769, A243947.
%Y A300945 Cf. A300946.
%K A300945 nonn,tabl
%O A300945 0,5
%A A300945 _Peter Luschny_, Mar 16 2018