login
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).
1

%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