A143330 G.f. satisfies: A(x) = (1 + x*A(x)^2)/(1 - x^2).

1, 1, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047, 76459765772375

Offset

0,3

Comments

Diagonal sums of A060693. - Paul Barry, Feb 11 2009

Starting with the second 1 and inserting a 2 between the 1 and 3: (1, 2, 3, 8, 25, 83, ...) the INVERT transform of that sequence deletes the 2, getting (1, 3, 8, 25, 83, ...). - Gary W. Adamson, Jun 24 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.

J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From N. J. A. Sloane, Dec 27 2012

Formula

G.f.: A(x) = (1-x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).

From Paul Barry, Feb 11 2009: (Start)

G.f.: 1/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-...))))) (continued fraction).

a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,k)*A000108(n-2k).

(End)

D-finite with recurrence (n+1)*a(n) +(n+2)*a(n-1) +2*(17-11n)*a(n-2) +10*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Dec 11 2011

a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 4.439109106851354261627... is the root of the equation 1 - 2*d^2 - 4*d^3 + d^4 = 0 and c = 1/2*sqrt(d*(d^2+3)/(d^2-1)) = 1.16064231... - Vaclav Kotesovec, Feb 03 2014

G.f. satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*x^k*A(x)^(j-k). - Ilya Gutkovskiy, Apr 11 2019

G.f.: 1/G(x), with G(x) = 1-(x+x^2)/(1-x/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 11 2023

From Peter Luschny, Jan 25 2023: (Start)

a(n) = CatalanNumber(n)*hypergeom([-n/2, -n/2, -n/2 - 1/2, -n/2 + 1/2], [-(2*n)/3, -(2*n)/3 + 1/3, -(2*n)/3 + 2/3], -16/27).

a(n) = ((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) for n >= 4. (End)

Example

G.f. = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + 1041*x^7 + ...

Maple

a := proc(n) option remember; if n <= 3 then return [1, 1, 3, 8][n + 1] fi;
((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) end:
seq(a(n), n = 0..25); # Peter Luschny, Jan 25 2023

Mathematica

CoefficientList[Series[(1 - x^2 - Sqrt[1 - 4 x - 2 x^2 + x^4])/(2 x), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 17 2013 *)

Prog

(PARI) {a(n)=polcoeff((1-x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x), n)}

Crossrefs

Cf. A000108, A060693.

Sequence in context: A258466 A216640 A148794 * A148795 A148796 A148797

Adjacent sequences: A143327 A143328 A143329 * A143331 A143332 A143333

Keyword

nonn

Author

Paul D. Hanna, Aug 08 2008

Extensions

Minor edits by Vaclav Kotesovec, Mar 31 2014

Status

approved