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A119372 G.f. satisfies: A(x) = 1 + x*(1-x-x^2)*A(x) + x^2*(3+2*x)*A(x)^2.

%I A119372 #15 Mar 18 2021 11:29:11
%S A119372 1,1,3,9,30,104,374,1380,5197,19893,77170,302716,1198729,4785455,
%T A119372 19238706,77821522,316506253,1293489529,5309112257,21876225899,
%U A119372 90459484106,375256749620,1561259497099,6513108751281,27238006266620
%N A119372 G.f. satisfies: A(x) = 1 + x*(1-x-x^2)*A(x) + x^2*(3+2*x)*A(x)^2.
%C A119372 Equals diagonal and row sums of pendular trinomial triangle A119369. Also equals convolution of A119370 and A119371 (central terms of A119369).
%H A119372 G. C. Greubel, <a href="/A119372/b119372.txt">Table of n, a(n) for n = 0..1000</a>
%F A119372 G.f.: A(x) = (1-x+x^2+x^3 - sqrt( (1-x+x^2+x^3)^2 - 4*x^2*(3+2*x)) )/(2*x^2*(3+2*x)).
%F A119372 G.f.: A(x) = B(x)/(1+x - x*B(x)) = B(x)*G(x), where B(x) is g.f. of A119370, G(x) is g.f. of A119371.
%F A119372 Recurrence: 3*(n+2)*(2*n-1)*a(n) = (20*n^2 - 6*n - 11)*a(n-1) + (28*n^2 - 18*n + 5)*a(n-2) + (8*n^2-12*n-17)*a(n-3) - 3*(2*n^2 - 9*n + 1)*a(n-4) - 2*(n-5)*(2*n+1)*a(n-5). - _Vaclav Kotesovec_, Sep 11 2013
%F A119372 a(n) ~ sqrt(-8*z^2-5*z^3+2-5*z)*(4+2*z-z^3)^n*(-18-8*z+4*z^3+z^2)*(-35+8*z^3-12*z^2+2*z)/(242*sqrt(Pi)*n^(3/2)), where z = 1/(2*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) - 1/2*sqrt(8/3-1/3*(280-24*sqrt(129))^(1/3) - 2/3*(35+3*sqrt(129))^(1/3) + 8*sqrt(3/(4+(280-24*sqrt(129))^(1/3) + 2*(35+3*sqrt(129))^(1/3)))) = 0.225270426... is the root of the equation 1-2*z^2+z^4-4*z=0. - _Vaclav Kotesovec_, Sep 11 2013
%p A119372 m:= 30;
%p A119372 S:= series( (1-x+x^2+x^3 -(1+x)*sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)), x, m+1);
%p A119372 seq(coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Mar 17 2021
%t A119372 CoefficientList[Series[(1-x+x^2+x^3-Sqrt[(1-x+x^2+x^3)^2-4*x^2*(3+2*x)])/(2*x^2*(3+2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%o A119372 (PARI) {a(n)=polcoeff(2/(1-x+x^2+x^3+sqrt((1-x+x^2+x^3)^2-4*x^2*(3+2*x)+x*O(x^n))),n)}
%o A119372 (Sage)
%o A119372 def A119372_list(prec):
%o A119372     P.<x> = PowerSeriesRing(QQ, prec)
%o A119372     return P( (1-x+x^2+x^3 -(1+x)*sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) ).list()
%o A119372 A119372_list(30) # _G. C. Greubel_, Mar 17 2021
%o A119372 (Magma)
%o A119372 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A119372 Coefficients(R!( (1-x+x^2+x^3 -(1+x)*Sqrt(1-4*x-2*x^2+x^4))/(2*x^2*(3+2*x)) )); // _G. C. Greubel_, Mar 17 2021
%Y A119372 Cf. A119369, A119370, A119371, A119373, A119374, A119375, A119376.
%K A119372 nonn
%O A119372 0,3
%A A119372 _Paul D. Hanna_, May 17 2006