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A247507 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).

%I A247507 #14 Jun 25 2021 05:51:00
%S A247507 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 A247507 132,1,7,42,205,740,1686,1806,429,1,8,56,330,1530,5028,9912,8558,1430,
%U A247507 1,9,72,497,2814,12130,35700,60213,41586,4862
%N A247507 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 A247507 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 A247507 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 A247507    [0][1] [2]  [3]    [4]     [5]      [6]       [7]
%e A247507 [0] 1, 1,  2,   5,    14,     42,     132,      429,.. A000108
%e A247507 [1] 1, 2,  6,  22,    90,    394,    1806,     8558,.. A006318
%e A247507 [2] 1, 3, 12,  57,   300,   1686,    9912,    60213,.. A047891
%e A247507 [3] 1, 4, 20, 116,   740,   5028,   35700,   261780,.. A082298
%e A247507 [4] 1, 5, 30, 205,  1530,  12130,  100380,   857405,.. A082301
%e A247507 [5] 1, 6, 42, 330,  2814,  25422,  239442,  2326434,.. A082302
%e A247507 [6] 1, 7, 56, 497,  4760,  48174,  507696,  5516133,.. A082305
%e A247507 [7] 1, 8, 72, 712,  7560,  84616,  985032, 11814728,.. A082366
%e A247507 [8] 1, 9, 90, 981, 11430, 140058, 1782900, 23369805,.. A082367
%p A247507 gf := n -> (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x):
%p A247507 for n from 0 to 10 do lprint(PolynomialTools:-CoefficientList( convert(series(gf(n),x,8),polynom),x)) od;
%Y A247507 Cf. A243631.
%Y A247507 Main diagonal gives A302286.
%K A247507 nonn,tabl
%O A247507 0,5
%A A247507 _Peter Luschny_, Nov 17 2014
%E A247507 Offset changed to 0 by _Alois P. Heinz_, May 28 2015