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
Number of Schroeder-like (see A006318) excursions (paths on or above height 0 beginning and ending at height 0) of semilength n, with steps U=(1,1), D=(1,-1), and H=(4,0). - Alexander Burstein, May 21 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, all internal nodes have weight 1, and leaf nodes have weights in {1,2}. - John Tyler Rascoe, Jun 06 2025
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Joseph P. S. Kung and Anna 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)
G.f. A(x) = -x + (1/x)*series_reversion(x*G(-x)), where G(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ... is the g.f. of A073157. - Peter Bala, Aug 27 2024
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
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Aug 08 2008
EXTENSIONS
Minor edits by Vaclav Kotesovec, Mar 31 2014
STATUS
approved