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

%I A143330 #76 Jun 07 2025 08:17:36
%S A143330 1,1,3,8,25,83,289,1041,3847,14504,55569,215727,846761,3354858,
%T A143330 13398965,53888063,218053915,887107888,3626373205,14887942624,
%U A143330 61358959587,253771944529,1052917272543,4381374717994,18280470530047,76459765772375
%N A143330 G.f. satisfies: A(x) = (1 + x*A(x)^2)/(1 - x^2).
%C A143330 Diagonal sums of A060693. - _Paul Barry_, Feb 11 2009
%C A143330 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
%C A143330 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
%C A143330 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
%H A143330 Vincenzo Librandi, <a href="/A143330/b143330.txt">Table of n, a(n) for n = 0..1000</a>
%H A143330 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
%H A143330 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A143330 Joseph P. S. Kung and Anna de Mier, <a href="http://dx.doi.org/10.1016/j.jcta.2012.08.010">Catalan lattice paths with rook, bishop and spider steps</a>, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - From _N. J. A. Sloane_, Dec 27 2012
%F A143330 G.f.: A(x) = (1-x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).
%F A143330 From _Paul Barry_, Feb 11 2009: (Start)
%F A143330 G.f.: 1/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-x^2-x/(1-...))))) (continued fraction).
%F A143330 a(n) = Sum_{k=0..floor(n/2)} C(2n-3k,k)*A000108(n-2k). (End)
%F A143330 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
%F A143330 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
%F A143330 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
%F A143330 G.f.: 1/G(x), with G(x) = 1-(x+x^2)/(1-x/G(x)) (continued fraction). - _Nikolaos Pantelidis_, Jan 11 2023
%F A143330 From _Peter Luschny_, Jan 25 2023: (Start)
%F A143330 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).
%F A143330 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)
%F A143330 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
%e A143330 G.f. = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 + 1041*x^7 + ...
%p A143330 a := proc(n) option remember; if n <= 3 then return [1, 1, 3, 8][n + 1] fi;
%p A143330 ((5 - n)*a(n - 4) + (2*n - 4)*a(n - 2) + (4*n - 2)*a(n - 1))/(n + 1) end:
%p A143330 seq(a(n), n = 0..25); # _Peter Luschny_, Jan 25 2023
%t A143330 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 *)
%o A143330 (PARI) {a(n)=polcoeff((1-x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x),n)}
%Y A143330 Cf. A000108, A006318, A060693, A073157.
%K A143330 nonn,easy
%O A143330 0,3
%A A143330 _Paul D. Hanna_, Aug 08 2008
%E A143330 Minor edits by _Vaclav Kotesovec_, Mar 31 2014