A357012 Triangle read by rows. T(n, k) = [x^k](0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1)).
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, 224, 168, 132, 0, 4096, 2048, 1280, 896, 672, 528, 429, 0, 16384, 8192, 5120, 3584, 2688, 2112, 1716, 1430, 0, 65536, 32768, 20480, 14336, 10752, 8448, 6864, 5720, 4862
Offset: 0
Examples
[0] 1; [1] 0, 1; [2] 0, 4, 2; [3] 0, 16, 8, 5; [4] 0, 64, 32, 20, 14; [5] 0, 256, 128, 80, 56, 42; [6] 0, 1024, 512, 320, 224, 168, 132; [7] 0, 4096, 2048, 1280, 896, 672, 528, 429; [8] 0, 16384, 8192, 5120, 3584, 2688, 2112, 1716, 1430; [9] 0, 65536, 32768, 20480, 14336, 10752, 8448, 6864, 5720, 4862;
Programs
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Maple
ogf := n -> 0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1): ser := n -> series(ogf(n), x, 32): seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..n)), n = 0..9);
Formula
T(n, 0) = 0^n, T(n, n) = CatalanNumber(n), otherwise T(n, k) = 4^(n - k)*T(k, k).