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A357012 Triangle read by rows. T(n, k) = [x^k](0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1)).

%I A357012 #4 Sep 09 2022 04:05:45
%S A357012 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 A357012 224,168,132,0,4096,2048,1280,896,672,528,429,0,16384,8192,5120,3584,
%U A357012 2688,2112,1716,1430,0,65536,32768,20480,14336,10752,8448,6864,5720,4862
%N A357012 Triangle read by rows. T(n, k) = [x^k](0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1)).
%F A357012 T(n, 0) = 0^n, T(n, n) = CatalanNumber(n), otherwise T(n, k) = 4^(n - k)*T(k, k).
%e A357012 [0] 1;
%e A357012 [1] 0,     1;
%e A357012 [2] 0,     4,     2;
%e A357012 [3] 0,    16,     8,     5;
%e A357012 [4] 0,    64,    32,    20,    14;
%e A357012 [5] 0,   256,   128,    80,    56,    42;
%e A357012 [6] 0,  1024,   512,   320,   224,   168,  132;
%e A357012 [7] 0,  4096,  2048,  1280,   896,   672,  528,  429;
%e A357012 [8] 0, 16384,  8192,  5120,  3584,  2688, 2112, 1716, 1430;
%e A357012 [9] 0, 65536, 32768, 20480, 14336, 10752, 8448, 6864, 5720, 4862;
%p A357012 ogf := n -> 0^n + 4^n * ((2 - 2*(1 - x)^(1/2)) / x - 1):
%p A357012 ser := n -> series(ogf(n), x, 32):
%p A357012 seq(lprint([n], seq(coeff(ser(n), x, k), k = 0..n)), n = 0..9);
%Y A357012 Cf. A000108, A000302, A008549 (row sums), A356651.
%K A357012 nonn,tabl
%O A357012 0,5
%A A357012 _Peter Luschny_, Sep 09 2022