A247507 - OEIS

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

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

L. Yang, S.-L. Yang, A relation between Schroder paths and Motzkin paths, Graphs Combinat. 36 (2020) 1489-1502, eq. (5).

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

EXAMPLE

[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

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;

CROSSREFS

Cf. A243631.

Main diagonal gives A302286.

Sequence in context: A153199 A056860 A158825 * A107111 A082037 A163649

Adjacent sequences: A247504 A247505 A247506 * A247508 A247509 A247510

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Nov 17 2014

EXTENSIONS

Offset changed to 0 by Alois P. Heinz, May 28 2015

STATUS

approved