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A143926 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)^2*A(-x)^2.

%I A143926 #12 Oct 22 2016 12:41:00
%S A143926 1,1,1,1,2,3,7,11,28,46,123,207,572,979,2769,4797,13806,24138,70414,
%T A143926 123998,365636,647615,1926505,3428493,10273870,18356714,55349155,
%U A143926 99229015,300783420,540807165,1646828655,2968468275,9075674700
%N A143926 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)^2*A(-x)^2.
%C A143926 Bisections form A006605 and A143927;
%C A143926 A006605 is the number of modes of connections of 2n points and
%C A143926 A143927 is the self-convolution of A006605.
%H A143926 Vaclav Kotesovec, <a href="/A143926/b143926.txt">Table of n, a(n) for n = 0..100</a>
%F A143926 Define B(x) by B(x^2) = A(x)*A(-x); then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4 is the g.f. of A006605.
%F A143926 Recurrence: 3*n*(3*n+1)*(3*n+2)*(507*n^4 - 3575*n^3 + 8895*n^2 - 8953*n + 3054)*a(n) =  - 12*(4017*n^5 - 20319*n^4 + 31895*n^3 - 17595*n^2 + 2338*n + 384)*a(n-1) + 4*(n-2)*(17745*n^6 - 125125*n^5 + 331891*n^4 - 396335*n^3 + 173912*n^2 + 17532*n - 13140)*a(n-2) - 144*(n-3)*(n-2)*(312*n^3 - 988*n^2 + 407*n + 29)*a(n-3) + 144*(n-4)*(n-3)*(n-2)*(507*n^4 - 1547*n^3 + 1212*n^2 + 140*n - 72)*a(n-4). - _Vaclav Kotesovec_, Dec 21 2013
%F A143926 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt(70/27+(26*sqrt(13))/27) = 2.4626418602647616787... is the root of the equation -144 - 140*d^2 + 27*d^4 = 0 and c = 2*sqrt((5+1/sqrt(13))/3)/3 = 0.88421131194123... if n is even, and c = sqrt(1+11/sqrt(13))/3 = 0.670890873659690... if n is odd. - _Vaclav Kotesovec_, Dec 21 2013
%e A143926 G.f. A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 11*x^7 +...
%e A143926 A(x)*A(-x) = 1 + x^2 + 3*x^4 + 11*x^6 + 46*x^8 + 207*x^10 + 979*x^12 +...
%e A143926 A(x)^2*A(-x)^2 = 1 + 2*x^2 + 7*x^4 + 28*x^6 + 123*x^8 + 572*x^10 +...
%e A143926 A(x)^4*A(-x)^4 = 1 + 4*x^2 + 18*x^4 + 84*x^6 + 407*x^8 + 2028*x^10 +...
%e A143926 from this we see that if B(x^2) = A(x)*A(-x)
%e A143926 then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4
%e A143926 and A(x) = 1 + x*B(x^2) + x^2*B(x^2)^2.
%t A143926 a[n_] := Module[{A = 1 + x, B}, For[i = 0, i <= n, i++, B = A*(A /. x -> -x); A = 1 + x*B + x^2*B^2 + O[x]^(n+1) // Normal]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Oct 22 2016, adapted from PARI *)
%o A143926 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,B=A*subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)}
%Y A143926 Cf. A006605, A143927.
%K A143926 nonn
%O A143926 0,5
%A A143926 _Paul D. Hanna_, Sep 06 2008