cp's OEIS Frontend

A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

%I A087626 #15 Mar 07 2023 11:06:24
%S A087626 1,2,5,13,36,104,311,955,2995,9553,30896,101082,333946,1112496,
%T A087626 3732955,12605029,42800317,146046819,500555447,1722402303,5948047169,
%U A087626 20607691517,71610355540,249520257106,871614139396,3051737703526
%N A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).
%H A087626 Robert Israel, <a href="/A087626/b087626.txt">Table of n, a(n) for n = 0..1762</a>
%H A087626 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%F A087626 G.f.: 2/(1-2x+sqrt(1-4x+4x^3)).
%F A087626 G.f. A(x) satisfies 0 = x^2*(1-x)*A(x)^2 - (1-2*x)*A(x) + 1.
%F A087626 First backwards difference is A056010.
%F A087626 (6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4)=0. - _Robert Israel_, Oct 26 2018
%F A087626 HANKEL transform is A006720(n+2). HANKEL transform with 0 prepended is -A051138.
%F A087626 INVERT transform of A157003. - _Michael Somos_, Mar 28 2020
%e A087626 G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 36*x^4 + 104*x^5 + 311*x^6 + 955*x^7 + ... - _Michael Somos_, Mar 28 2020
%p A087626 f:= gfun:-rectoproc({(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 13},a(n),remember):
%p A087626 map(f, [$0..50]); # _Robert Israel_, Oct 26 2018
%t A087626 CoefficientList[Series[2/(1-2x+Sqrt[1-4x+4x^3]),{x,0,30}],x] (* _Harvey P. Dale_, Jun 12 2017 *)
%o A087626 (PARI) {a(n) = polcoeff(2 / (1 - 2*x + sqrt(1 - 4*x + 4*x^3 + x*O(x^n))), n)};
%Y A087626 Cf. A006720, A051138, A056010, A157003.
%K A087626 nonn
%O A087626 0,2
%A A087626 _Michael Somos_, Sep 16 2003