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A254316 Hankel transform of a(n) is A006720(n+1). Hankel transform of a(n+1) is A006720(n+3).

%I A254316 #11 Aug 04 2018 02:39:56
%S A254316 1,1,2,6,21,78,299,1172,4677,18947,77746,322545,1350906,5704822,
%T A254316 24265651,103872254,447146683,1934538301,8407277728,36685185300,
%U A254316 160663301053,705974374128,3111584887543,13752592535137,60939737103636,270672216346769,1204862348053296
%N A254316 Hankel transform of a(n) is A006720(n+1). Hankel transform of a(n+1) is A006720(n+3).
%H A254316 G. C. Greubel, <a href="/A254316/b254316.txt">Table of n, a(n) for n = 0..1000</a>
%F A254316 Given g.f. A(x), 0 = (x^2-x)*A(x)^2 + (x^2-2*x+1)*A(x) + (2*x-1).
%F A254316 G.f.: (1 - 2*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*x*(1 - x)).
%F A254316 Conjecture: +(n+1)*a(n) +(-8*n+3)*a(n-1) +(18*n-29)*a(n-2) +(-12*n+31)*a(n-3) +(n-4)*a(n-4)=0. - _R. J. Mathar_, Jun 07 2016
%e A254316 G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 299*x^6 + 1172*x^7 + ...
%t A254316 CoefficientList[Series[(1-2*x+x^2-Sqrt[(1-4*x+x^2)^2-4*x^3])/(2*x*(1 - x)), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 04 2018 *)
%o A254316 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 2*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 + x^2 * O(x^n))) / (2*x*(1 - x)), n))};
%Y A254316 Cf. A006720.
%K A254316 nonn
%O A254316 0,3
%A A254316 _Michael Somos_, Jan 28 2015