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%I #40 Jan 25 2024 12:35:11
%S 1,2,5,12,30,76,197,520,1398,3820,10594,29768,84620,243000,704045,
%T 2055760,6043750,17875020,53148310,158773320,476311940,1434313960,
%U 4333867170,13135533552,39924668220,121661345656,371612931492
%N G.f. A(x) satisfies: A(x) = 1/(1-2*x) + x^2*A(x)^2.
%C a(n) is the number of Motzkin paths of length n having no (1,0)-steps at levels 1,3,5,... and having (1,0)-steps of two colors at levels 2,4,6,... . Example: a(3) = 12 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 8 paths of shape HHH, 2 paths of shape HUD, and 2 paths of shape UDH. - _Emeric Deutsch_, May 02 2011
%H Vincenzo Librandi, <a href="/A086622/b086622.txt">Table of n, a(n) for n = 0..200</a>
%H Carles Cardó, <a href="https://arxiv.org/abs/2401.07827">Growth and density in free magmas</a>, arXiv:2401.07827 [math.CO], 2024. See p. 16.
%F Antidiagonal sums of square table A086620.
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k) C(2k,k) 2^(n-2k)/(k+1). - _Paul Barry_, Nov 13 2004
%F Hankel transform of a(n) is 1,1,1,....; Hankel transform of a(n+1) is A009531(n+2). - _Paul Barry_, Nov 06 2007
%F G.f.: 1/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/..... (continued fraction). - _Paul Barry_, Dec 21 2008
%F D-finite with recurrence (n+2)*a(n) +4*(-n-1)*a(n-1) +4*a(n-2) +4*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012
%F G.f.: (-1+2*x+sqrt(1-4*x+8*x^3))/(2*(-x^2+2*x^3)). - _Vaclav Kotesovec_, Feb 13 2014
%F a(n) ~ sqrt(50+22*sqrt(5)) * (sqrt(5)+1)^n / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014. Equivalently, a(n) ~ 5^(1/4) * 2^n * phi^(n + 5/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 07 2021
%F a(n) = Sum_{i=0..floor(n/2)}2^(n-2i)*C(i)*binomial(n-i,i), where C(n) is the n-th Catalan number A000108. - _José Luis Ramírez Ramírez_, Apr 20 2015
%p A086622 := proc(n)
%p option remember;
%p if n < 3 then
%p op(n+1,[1,2,5]) ;
%p else
%p 4*(-n-1)*procname(n-1) +4*procname(n-2) +4*(2*n-3)*procname(n-3) ;
%p -%/(n+2) ;
%p end if;
%p end proc:
%p seq(A086622(n),n=0..20) ; # _R. J. Mathar_, Nov 02 2021
%t CoefficientList[Series[(-1+2*x+Sqrt[1-4*x+8*x^3])/(2*(-x^2+2*x^3)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y Cf. A086620 (table), A086621 (diagonal).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 24 2003
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