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).
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, 132, 1, 7, 42, 205, 740, 1686, 1806, 429, 1, 8, 56, 330, 1530, 5028, 9912, 8558, 1430, 1, 9, 72, 497, 2814, 12130, 35700, 60213, 41586, 4862
Offset: 0
Examples
[0][1] [2] [3] [4] [5] [6] [7] [0] 1, 1, 2, 5, 14, 42, 132, 429,.. A000108 [1] 1, 2, 6, 22, 90, 394, 1806, 8558,.. A006318 [2] 1, 3, 12, 57, 300, 1686, 9912, 60213,.. A047891 [3] 1, 4, 20, 116, 740, 5028, 35700, 261780,.. A082298 [4] 1, 5, 30, 205, 1530, 12130, 100380, 857405,.. A082301 [5] 1, 6, 42, 330, 2814, 25422, 239442, 2326434,.. A082302 [6] 1, 7, 56, 497, 4760, 48174, 507696, 5516133,.. A082305 [7] 1, 8, 72, 712, 7560, 84616, 985032, 11814728,.. A082366 [8] 1, 9, 90, 981, 11430, 140058, 1782900, 23369805,.. A082367
Links
- L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (5).
Programs
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Maple
gf := n -> (1-n*x-sqrt(n^2*x^2-2*n*x-4*x+1))/(2*x): for n from 0 to 10 do lprint(PolynomialTools:-CoefficientList( convert(series(gf(n),x,8),polynom),x)) od;
Formula
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
Extensions
Offset changed to 0 by Alois P. Heinz, May 28 2015