cp's OEIS Frontend

A119371 G.f. satisfies: A(x) = (1+x) - x*(4+x)*A(x) + x*(3+2*x)*A(x)^2.

%I A119371 #15 Mar 18 2021 11:29:04
%S A119371 1,0,1,2,7,23,81,293,1087,4110,15783,61387,241329,957400,3828055,
%T A119371 15410651,62410871,254095382,1039394147,4269718110,17606507789,
%U A119371 72852976317,302403773303,1258855723796,5254253027485,21983753239135
%N A119371 G.f. satisfies: A(x) = (1+x) - x*(4+x)*A(x) + x*(3+2*x)*A(x)^2.
%C A119371 Equals central terms of pendular trinomial triangle A119369.
%H A119371 G. C. Greubel, <a href="/A119371/b119371.txt">Table of n, a(n) for n = 0..1000</a>
%F A119371 G.f.: A(x) = (1+4*x+x^2 - sqrt((1+4*x+x^2)^2 - 4*x*(1+x)*(3+2*x)))/(2*x*(3+2*x)).
%F A119371 G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A119370, H(x) is g.f. of A119372, F(x) is g.f. of A119375.
%F A119371 Recurrence: 3*(n+1)*a(n) = 2*(5*n-4)*a(n-1) + 2*(7*n-8)*a(n-2) + 4*(n-2)*a(n-3) - 3*(n-5)*a(n-4) - 2*(n-5)*a(n-5). - _Vaclav Kotesovec_, Sep 11 2013
%F A119371 a(n) ~ sqrt(-z^2-3*z+1)*(4+2*z-z^3)^(n+1)*(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
%e A119371             A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 23*x^5 + 81*x^6 ...;
%e A119371    -x*(4+x)*A(x) = -4*x -x^2 -4*x^3 -9*x^4 -30*x^5 -99*x^6 - ...;
%e A119371 x*(3+2*x)*A(x)^2 = 3*x +2*x^2 +6*x^3 +16*x^4 +53*x^5 +180*x^6 + ...;
%p A119371 m:= 30;
%p A119371 S:= series( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)), x, m+1);
%p A119371 seq(coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Mar 17 2021
%t A119371 CoefficientList[Series[(1+4*x+x^2-Sqrt[(1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)])/(2*x*(3+2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%o A119371 (PARI) {a(n)=polcoeff((2*(1+x))/(1+4*x+x^2+sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x*O(x^n))),n)}
%o A119371 (Sage)
%o A119371 def A119371_list(prec):
%o A119371     P.<x> = PowerSeriesRing(QQ, prec)
%o A119371     return P( (1+4*x+x^2 -sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) ).list()
%o A119371 A119371_list(30) # _G. C. Greubel_, Mar 17 2021
%o A119371 (Magma)
%o A119371 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A119371 Coefficients(R!( (1+4*x+x^2 -Sqrt(1-4*x-2*x^2+x^4))/(2*x*(3+2*x)) )); // _G. C. Greubel_, Mar 17 2021
%Y A119371 Cf. A119369, A119370, A119372, A119373, A119374, A119375, A119376.
%K A119371 nonn
%O A119371 0,4
%A A119371 _Paul D. Hanna_, May 17 2006