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G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 8*x.
3

%I #8 Aug 31 2013 00:33:05

%S 1,2,-5,22,-115,646,-3822,23496,-148368,955822,-6256273,41480668,

%T -277954706,1879118354,-12800031737,87758481546,-605091552753,

%U 4192829686338,-29180958305391,203887504096188,-1429568781831693,10055261467844862,-70929518958227340

%N G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 8*x.

%F G.f. A(x) satisfies:

%F (1) A(x) = sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2),

%F (2) sqrt(A(x^2)^2 + 4*x) = 1/G(x^4) + 2*x*G(x^4),

%F where G(x) is the g.f. of A228712.

%F Self-convolution of A223026.

%e G.f.: A(x) = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +...

%e where A(x)^4 = A(x^2)^2 + 8*x as demonstrated by:

%e A(x)^2 = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 +...

%e A(x)^4 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +...

%e The g.f. of A228712 begins:

%e G(x) = 1 + 3*x + 72*x^2 + 2307*x^3 + 86295*x^4 + 3513477*x^5 +...

%e and satisfies: sqrt(1/G(x^2)^2 + 4*x*G(x^2)^2) = A(x).

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+8*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A228712, A107086, A223026.

%K sign

%O 0,2

%A _Paul D. Hanna_, Aug 30 2013