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

Author

Peter Luschny, Nov 17 2014

Keywords

• nonn
• tabl

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).

Crossrefs

Cf. A243631.

Main diagonal gives A302286.

Programs

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