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A182959 Expansion of o.g.f. 2*(1+x)^2/(1-2*x+sqrt(1-8*x)).

%I A182959 #37 Nov 11 2024 22:36:41
%S A182959 1,5,20,96,528,3136,19584,126720,841984,5710848,39376896,275185664,
%T A182959 1944821760,13875707904,99807723520,722997411840,5269761884160,
%U A182959 38620004352000,284405842575360,2103530005463040,15619068033761280
%N A182959 Expansion of o.g.f. 2*(1+x)^2/(1-2*x+sqrt(1-8*x)).
%H A182959 G. C. Greubel, <a href="/A182959/b182959.txt">Table of n, a(n) for n = 0..1000</a>
%F A182959 Let F(x) be the g.f. of A182960, then g.f. of this sequence satisfies:
%F A182959 * A(x) = F(x/A(x)^3) and A(x*F(x)^3) = F(x);
%F A182959 * A(x) = [x/Series_Reversion( x*F(x)^3 )]^(1/3).
%F A182959 G.f.: 1/2/x - 1/2 - x - (1+x)/x/G(0), where G(k)= 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F A182959 a(n) ~ 9*2^(3*n-2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013
%F A182959 From _Peter Bala_, Oct 04 2015: (Start)
%F A182959 O.g.f. A(x) = (1 + x)*(2*C(2*x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
%F A182959 [x^n] A(x)^(3*n) = binomial(6*n,2*n). Cf. with the identity [x^n] ( (1 + x)*C(x) )^(5*n) = binomial(5*n,2*n) = A001450(n). (End)
%F A182959 Conjecture: D-finite with recurrence (n+1)*a(n) +(-7*n+3)*a(n-1) +4*(-2*n+5)*a(n-2)=0. - _R. J. Mathar_, Jan 22 2020
%F A182959 From _Peter Bala_, May 15 2023: (Start)
%F A182959 a(n) = 3*(2^n)*(3*n - 1)/(n*(n + 1)) * binomial(2*n-2,n-1) for n >= 2.
%F A182959 (n + 1)*(3*n - 4)*a(n) = 4*(2*n - 3)*(3*n - 1)*a(n-1) for n >= 3 with a(2) = 20. Mathar's conjectured second order recurrence above follows from this. (End)
%F A182959 [x^n] A(x)^n = A372215(n). - _Peter Bala_, Nov 07 2024
%e A182959 G.f.: A(x) = 1 + 5*x + 20*x^2 + 96*x^3 + 528*x^4 + 3136*x^5 +...
%e A182959 where A(x*F(x)^3) = F(x) is the g.f. of A182960:
%e A182959 F(x) = 1 + 5*x + 95*x^2 + 2496*x^3 + 76063*x^4 + 2524161*x^5 +...
%t A182959 CoefficientList[ Series[2 (1 + x)^2/(1 - 2 x + Sqrt[1 - 8 x]), {x, 0, 20}], x]  (* _Robert G. Wilson v_, Dec 31 2010 *)
%o A182959 (PARI) {a(n)=polcoeff(2*(1+x)^2/(1-2*x+sqrt(1-8*x+x*O(x^n))),n)}
%Y A182959 Cf. A182960, A001450, A167422, A372215.
%K A182959 nonn,easy
%O A182959 0,2
%A A182959 _Paul D. Hanna_, Dec 31 2010