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A143547 G.f. satisfies: A(x) = 1 + x*A(x)^4*A(-x)^3. 9

%I

%S 1,1,1,4,7,34,70,368,819,4495,10472,59052,141778,814506,1997688,

%T 11633440,28989675,170574723,430321633,2552698720,6503352856,

%U 38832808586,99726673130,598724403680,1547847846090,9335085772194

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

%H Michel Bousquet and Cédric Lamathe, <a href="http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/1003">On symmetric structures of order two</a>, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176, See Table 1. - From _N. J. A. Sloane_, Jul 12 2011

%F G.f.: A(x) = G(x^2) + x*G(x^2)^4 where G(x^2) = A(x)*A(-x) and G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

%F a(2n) = C(7*n,n)/(6*n+1); a(2n+1) = C(7*n+3,n)*4/(6*n+4).

%F G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2.

%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 +...

%e Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then

%e A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where

%e G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...

%e G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 +...

%e form the bisections of A(x).

%e By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where

%e A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 +...

%e A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+...

%t terms = 26;

%t A[_] = 1; Do[A[x_] = 1 + x A[x]^4 A[-x]^3 + O[x]^terms // Normal, {terms}];

%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 24 2018 *)

%o (PARI) {a(n)=local(A=1+O(x^(n+1)));for(i=0,n,A=1+x*A^4*subst(A^3,x,-x));polcoeff(A,n)}

%o (PARI) {a(n)=local(m=n\2,p=3*(n%2)+1);binomial(7*m+p-1,m)*p/(6*m+p)}

%Y Cf. A002296 (bisection), A143546.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 23 2008

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Last modified May 16 23:34 EDT 2021. Contains 343957 sequences. (Running on oeis4.)