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%I #4 Sep 09 2022 04:05:45
%S 1,0,1,0,4,2,0,16,8,5,0,64,32,20,14,0,256,128,80,56,42,0,1024,512,320,
%T 224,168,132,0,4096,2048,1280,896,672,528,429,0,16384,8192,5120,3584,
%U 2688,2112,1716,1430,0,65536,32768,20480,14336,10752,8448,6864,5720,4862
%N Triangle read by rows. T(n, k) = [x^k](0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1)).
%F T(n, 0) = 0^n, T(n, n) = CatalanNumber(n), otherwise T(n, k) = 4^(n - k)*T(k, k).
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 4, 2;
%e [3] 0, 16, 8, 5;
%e [4] 0, 64, 32, 20, 14;
%e [5] 0, 256, 128, 80, 56, 42;
%e [6] 0, 1024, 512, 320, 224, 168, 132;
%e [7] 0, 4096, 2048, 1280, 896, 672, 528, 429;
%e [8] 0, 16384, 8192, 5120, 3584, 2688, 2112, 1716, 1430;
%e [9] 0, 65536, 32768, 20480, 14336, 10752, 8448, 6864, 5720, 4862;
%p ogf := n -> 0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1):
%p ser := n -> series(ogf(n), x, 32):
%p seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..n)), n = 0..9);
%Y Cf. A000108, A000302, A008549 (row sums), A356651.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Sep 09 2022