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%I #15 Nov 07 2019 05:25:41
%S 1,1,-1,2,-5,13,-35,99,-289,857,-2578,7864,-24252,75430,-236348,
%T 745431,-2364399,7536482,-24127482,77544613,-250098478,809169322,
%U -2625483810,8541037140,-27851360659,91018956200,-298052119611,977825373366,-3213513271929,10577811289462,-34870732260397
%N G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 4*x.
%C Self-convolution is A107087. Self-convolution 4th power is A107088.
%H Paul D. Hanna, <a href="/A107086/b107086.txt">Table of n, a(n) for n = 0..1000</a>
%e A(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
%e A(x^2)^2 = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
%t nmin = 0; nmax = 30; sol = {a[0] -> 1};
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x^2]^2 - 4x + O[x]^(n+1), x][[2;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 2, nmax}];
%t a /@ Range[nmin, nmax] /. sol (* _Jean-François Alcover_, Nov 07 2019 *)
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+4*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A107087, A107088, A187814, A228711.
%K sign
%O 0,4
%A _Paul D. Hanna_, May 11 2005
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