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A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.

%I A157004 #37 Mar 23 2023 11:45:07
%S A157004 1,2,6,18,58,192,650,2232,7746,27096,95376,337404,1198546,4272308,
%T A157004 15273888,54744268,196646922,707747988,2551624304,9213416524,
%U A157004 33313656888,120604436624,437112790668,1585877246424,5759085911154
%N A157004 Transform of central binomial coefficients A000984 whose Hankel transform obeys a Somos-4 recurrence.
%C A157004 Hankel transform is A157005. Image of A000984 under Riordan array (1,x(1-x^2)).
%C A157004 Diagonal of rational function 1/(1 - x - y + x^3*y^2). - _Seiichi Manyama_, Mar 23 2023
%H A157004 G. C. Greubel, <a href="/A157004/b157004.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%F A157004 G.f.: 1/sqrt(1 - 4*x*(1 - x^2)).
%F A157004 a(n) = Sum_{k=0..n} (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2)) *A000984(k)/2.
%F A157004 G.f.: Sum_{n>=0} (2*n)!/n!^2 * x^(2*n) * (1-x)^n / (1-2*x)^(2*n+1). - _Paul D. Hanna_, Sep 21 2013
%F A157004 D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) - 2*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Feb 13 2014
%F A157004 a(n) ~ (1/r)^n / (sqrt(Pi*n) * sqrt(3-8*r)), where r = 0.2695944364054... is the root of the equation 4*r*(1-r^2)=1. - _Vaclav Kotesovec_, Feb 13 2014
%F A157004 0 = a(n)*(16*a(n+1) - 32*a(n+3) + 10*a(n+4)) + a(n+1)*(-2*a(n+3)) + a(n+2)*(16*a(n+3) - 6*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - _Michael Somos_, Sep 03 2016
%e A157004 G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 192*x^5 + 650*x^6 + 2232*x^7 + ...
%t A157004 CoefficientList[Series[1/Sqrt[1-4*x*(1-x^2)], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o A157004 (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/m!^2 * x^(2*m)*(1-x)^m / (1-2*x+x*O(x^n))^(2*m+1)), n)} \\ _Paul D. Hanna_, Sep 21 2013
%o A157004 (PARI) my(x='x+O('x^30)); Vec(1/sqrt(1-4*x+4*x^3)) \\ _G. C. Greubel_, Feb 26 2019
%o A157004 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/Sqrt(1-4*x+4*x^3) )); // _G. C. Greubel_, Feb 26 2019
%o A157004 (Sage) (1/sqrt(1-4*x+4*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 26 2019
%Y A157004 Cf. A157003, A360219, A360266.
%K A157004 easy,nonn
%O A157004 0,2
%A A157004 _Paul Barry_, Feb 20 2009