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A122446 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 2*x^2*(A(x)^2 - A(x)); equals the base sequence of pendular trinomial triangle A122445.

%I A122446 #20 Mar 18 2024 05:34:58
%S A122446 1,1,2,7,24,88,336,1321,5316,21788,90640,381750,1624592,6975136,
%T A122446 30177056,131428917,575765820,2535433668,11216757104,49829385786,
%U A122446 222193501760,994153952528,4461915817760,20082611971226,90625360612296
%N A122446 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 2*x^2*(A(x)^2 - A(x)); equals the base sequence of pendular trinomial triangle A122445.
%C A122446 Functional equation for the g.f. is derived from the recurrence of the pendular triangle A122445. Iterated convolutions of this sequence with the central terms (A122447) generates all diagonals of A122445. For example: A122448 = A122446 * A122447; A122449 = A122446^2 * A122447.
%C A122446 Diagonal sums of triangle T with T(n,k) = 2^k*A133336(n,k). - _Philippe Deléham_, Nov 10 2009
%H A122446 G. C. Greubel, <a href="/A122446/b122446.txt">Table of n, a(n) for n = 0..1000</a>
%F A122446 G.f.: A(x) = (1 + 2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4))/(2*x*(1+2*x)).
%F A122446 Recurrence: (n+1)*a(n) = 2*(n-2)*a(n-1) + 12*(n-1)*a(n-2) + 8*(n-2)*a(n-3) - 4*(n-5)*a(n-4) - 8*(n-5)*a(n-5). - _Vaclav Kotesovec_, Sep 17 2013
%F A122446 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = (1/6)*(6+sqrt(6*(10 + 2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3))) + sqrt(6*(20-2^(2/3)*(43-3*sqrt(177))^(1/3) - 2^(2/3)*(43+3*sqrt(177))^(1/3) + 24*sqrt(6/(10+2^(2/3)*(43-3*sqrt(177))^(1/3) + 2^(2/3)*(43+3*sqrt(177))^(1/3)))))) = 4.797536514160165558... is the root of the equation 4 - 4*d^2 - 4*d^3 + d^4 = 0 and c = 0.908214882020417619380249683... is the positive root of the equation -59 - 944*c^2 - 2032*c^4 - 320*c^6 + 5184*c^8 = 0. - _Vaclav Kotesovec_, Sep 17 2013, updated Mar 18 2024
%p A122446 m:=30; S:=series( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)), x, m+1):
%p A122446 seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 16 2021
%t A122446 CoefficientList[Series[(1+2*x^2-Sqrt[1-4*x-4*x^2+4*x^4])/(2*x*(1+2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o A122446 (PARI) {a(n)=polcoeff(2/(1+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))),n)}
%o A122446 (Sage)
%o A122446 def A122446_list(prec):
%o A122446     P.<x> = PowerSeriesRing(QQ, prec)
%o A122446     return P( (1+2*x^2 -sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) ).list()
%o A122446 A122446_list(30) # _G. C. Greubel_, Mar 16 2021
%o A122446 (Magma)
%o A122446 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A122446 Coefficients(R!( (1+2*x^2 -Sqrt(1-4*x-4*x^2+4*x^4))/(2*x*(1+2*x)) )); // _G. C. Greubel_, Mar 16 2021
%Y A122446 Cf. A122445, A122447, A122448, A122449, A122450, A122451, A122452.
%K A122446 nonn
%O A122446 0,3
%A A122446 _Paul D. Hanna_, Sep 07 2006