A356651 Triangle read by rows. T(n, k) = [x^k](0^n + 4^n * ((1 - x)^(-1/2) - 1)).
1, 0, 2, 0, 8, 6, 0, 32, 24, 20, 0, 128, 96, 80, 70, 0, 512, 384, 320, 280, 252, 0, 2048, 1536, 1280, 1120, 1008, 924, 0, 8192, 6144, 5120, 4480, 4032, 3696, 3432, 0, 32768, 24576, 20480, 17920, 16128, 14784, 13728, 12870, 0, 131072, 98304, 81920, 71680, 64512, 59136, 54912, 51480, 48620
Offset: 0
Examples
[0] 1; [1] 0, 2; [2] 0, 8, 6; [3] 0, 32, 24, 20; [4] 0, 128, 96, 80, 70; [5] 0, 512, 384, 320, 280, 252; [6] 0, 2048, 1536, 1280, 1120, 1008, 924; [7] 0, 8192, 6144, 5120, 4480, 4032, 3696, 3432; [8] 0, 32768, 24576, 20480, 17920, 16128, 14784, 13728, 12870; [9] 0, 131072, 98304, 81920, 71680, 64512, 59136, 54912, 51480, 48620;
Programs
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Maple
ogf := n -> 0^n + 4^n * ((1 - x)^(-1/2) - 1): ser := n -> series(ogf(n), x, 32): seq(seq(coeff(ser(n), x, k), k = 0..n), n = 0..9);
Formula
T(n, 0) = 0^n, T(n, n) = binomial(2*n, n), otherwise T(n, k) = 4^(n - k)*T(k, k).