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%I #14 Jun 25 2021 05:51:00
%S 1,1,1,1,2,2,1,3,6,5,1,4,12,22,14,1,5,20,57,90,42,1,6,30,116,300,394,
%T 132,1,7,42,205,740,1686,1806,429,1,8,56,330,1530,5028,9912,8558,1430,
%U 1,9,72,497,2814,12130,35700,60213,41586,4862
%N Square array read by ascending antidiagonals, n>=0, k>=0. Row n is the expansion of (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x).
%H L. Yang, S.-L. Yang, <a href="https://doi.org/10.1007/s00373-020-02185-6">A relation between Schroder paths and Motzkin paths</a>, Graphs Combinat. 36 (2020) 1489-1502, eq. (5).
%F G.f. of row n: 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 06 2018
%e [0][1] [2] [3] [4] [5] [6] [7]
%e [0] 1, 1, 2, 5, 14, 42, 132, 429,.. A000108
%e [1] 1, 2, 6, 22, 90, 394, 1806, 8558,.. A006318
%e [2] 1, 3, 12, 57, 300, 1686, 9912, 60213,.. A047891
%e [3] 1, 4, 20, 116, 740, 5028, 35700, 261780,.. A082298
%e [4] 1, 5, 30, 205, 1530, 12130, 100380, 857405,.. A082301
%e [5] 1, 6, 42, 330, 2814, 25422, 239442, 2326434,.. A082302
%e [6] 1, 7, 56, 497, 4760, 48174, 507696, 5516133,.. A082305
%e [7] 1, 8, 72, 712, 7560, 84616, 985032, 11814728,.. A082366
%e [8] 1, 9, 90, 981, 11430, 140058, 1782900, 23369805,.. A082367
%p gf := n -> (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x):
%p for n from 0 to 10 do lprint(PolynomialTools:-CoefficientList( convert(series(gf(n),x,8),polynom),x)) od;
%Y Cf. A243631.
%Y Main diagonal gives A302286.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Nov 17 2014
%E Offset changed to 0 by _Alois P. Heinz_, May 28 2015
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