A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.
1, 0, 1, 0, 1, 2, 0, 2, 3, 3, 0, 5, 7, 7, 5, 0, 14, 19, 19, 15, 8, 0, 42, 56, 56, 46, 30, 13, 0, 132, 174, 174, 146, 103, 58, 21, 0, 429, 561, 561, 477, 351, 220, 109, 34, 0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55, 0, 4862, 6292, 6292, 5434, 4180, 2884, 1756, 908, 365, 89
Offset: 0
Examples
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 5, 14, 42, 132, 429, 1430, ... [A000108]
2, 3, 7, 19, 56, 174, 561, 1859, 6292, ... [A005807]
3, 7, 19, 56, 174, 561, 1859, 6292, 21658, ... [A005807]
5, 15, 46, 146, 477, 1595, 5434, 18798, 65858, ...
8, 30, 103, 351, 1205, 4180, 14651, 51844, 185028, ...
13, 58, 220, 801, 2884, 10372, 37401, 135420, 492558, ...
21, 109, 453, 1756, 6621, 24674, 91532, 339184, 1257762, ...
34, 201, 908, 3734, 14719, 56796, 216698, 821848, 3107583, ...
55, 365, 1781, 7746, 31872, 127245, 499164, 1937439, 7470819, ...
A000045,A023610,...
Seen as a triangle a refinement of A000958:
[0] 1
[1] 0, 1
[2] 0, 1, 2
[3] 0, 2, 3, 3
[4] 0, 5, 7, 7, 5
[5] 0, 14, 19, 19, 15, 8
[6] 0, 42, 56, 56, 46, 30, 13
[7] 0, 132, 174, 174, 146, 103, 58, 21
[8] 0, 429, 561, 561, 477, 351, 220, 109, 34
[9] 0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55
Crossrefs
Programs
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Maple
gf := 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1): serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20): row := n -> seq(coeff(sery(n), y, j), j=0..9): seq(lprint(row(n)), n=0..9);
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Mathematica
m = 11; T = PadRight[CoefficientList[#+O[y]^m, y], m]& /@ CoefficientList[1 + 2x(x+1)/(Sqrt[1-4y] - 2x(x+1) + 1) + O[x]^m, x]; Table[T[[n-k+1, k]], {n, 1, m}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)