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

Author

Peter Luschny, Sep 09 2022

Keywords

• nonn
• tabl

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;
		

Crossrefs

Cf. A000108, A000302, A008549 (row sums), A356651.

Programs

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).