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%I #34 Oct 02 2023 11:09:53
%S 1,2,10,58,370,2514,17850,130890,983650,7536418,58648810,462306266,
%T 3683602130,29620138994,240059315610,1958940281322,16081662931650,
%U 132723191430210,1100568370427850,9164925012016506,76612776253995570
%N Row sums of A059450.
%C Hankel transform is A165928. - _Paul Barry_, Sep 30 2009
%C Number of skew Dyck paths of semilength n with the down steps coming in two colors. - _David Scambler_, Jun 21 2013
%H Vincenzo Librandi, <a href="/A086871/b086871.txt">Table of n, a(n) for n = 0..200</a>
%H C. Coker, <a href="https://doi.org/10.1016/S0012-365X(03)00037-2">Enumerating a class of lattice paths</a>, Discrete Math., 271 (2003), 13-28.
%H J. Machacek, <a href="https://arxiv.org/abs/2105.02417">Lattice walks ending on a coordinate hyperlane avoiding backtracking and repeats</a>, arXiv:2105.02417 [math.CO], 2021. See Thm. 4.4 G(x,E^1).
%F a(n) = 2*A059231(n), if n>0.
%F G.f.: (1 - x - sqrt((1 - x) * (1 - 9*x))) / (4*x) = 2 / (1 + sqrt((1 - 9*x) / (1 - x))) =: y satisfies 0 = (1 - x) * (1 - y) + 2*x*y^2. - _Michael Somos_, Mar 06 2004
%F Moment representation: a(n) = (1/(4*Pi))*Integral_{x=1..9} x^n*sqrt(-x^2+10x-9)/x+(1/2)*0^n. - _Paul Barry_, Sep 30 2009
%F D-finite with recurrence Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F G.f.: 1/(2*x) - 1/2 + G(0) where G(k) = 1 - 1/(x + x/(1 + 1/G(k+1) )) ; (continued fraction,3-step). - _Sergei N. Gladkovskii_, Nov 29 2012
%e G.f. = 1 + 2*x + 10*x^2 + 58*x^3 + 370*x^4 + 2514*x^5 + 17850*x^6 + 130890*x^7 + ...
%t Table[SeriesCoefficient[2/(1+Sqrt[(1-9*x)/(1-x)]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (1 + sqrt((1 - 9*x) / (1 - x) + x * O(x^n))), n))}; /* _Michael Somos_, Mar 06 2004 */
%o (PARI) {a(n) = if( n<1, n==0, n++; 2 * polcoeff( serreverse( x * (1 - 4*x) / (1 - 3*x) + x * O(x^n)), n))}; /* _Michael Somos_, Mar 06 2004 */
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Sep 16 2003
%E More terms from _Ray Chandler_, Sep 17 2003
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