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A143549 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x).

%I A143549 #15 Jul 08 2025 07:41:51
%S A143549 1,1,3,17,85,598,3473,26668,166429,1340079,8724438,72374714,484498327,
%T A143549 4102336176,28009706440,240729330116,1668007246157,14499527706129,
%U A143549 101618389067849,891275643857227,6303425058175018,55686806813191060
%N A143549 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x).
%H A143549 Robert Israel, <a href="/A143549/b143549.txt">Table of n, a(n) for n = 0..600</a>
%F A143549 G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^5.
%F A143549 G.f. satisfies: -x^3*A(x)^15+x^2*A(x)^12-x^2*A(x)^11+A(x)^4-4*A(x)^3+6*A(x)^2-4*A(x)+1 = 0. - _Robert Israel_, Jul 10 2017
%F A143549 a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l) * a(m). - _Seiichi Manyama_, Jul 08 2025
%e A143549 G.f. A(x) = 1 + x + 3*x^2 + 17*x^3 + 85*x^4 + 598*x^5 + 3473*x^6 +...
%e A143549 Related expansions:
%e A143549 A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 635*x^4 + 4348*x^5 + 28336*x^6 +...
%e A143549 A(x)*A(-x) = 1 + 5*x^2 + 145*x^4 + 5971*x^6 + 287253*x^8 +...
%e A143549 [A(x)*A(-x)]^5 = 1 + 25*x^2 + 975*x^4 + 45605*x^6 + 2355490*x^8 +...
%p A143549 S:= series(RootOf(_Z^15*x^3-_Z^12*x^2+_Z^11*x^2-_Z^4+4*_Z^3-6*_Z^2+4*_Z-1),x,31):
%p A143549 seq(coeff(S,x,i),i=0..30); # _Robert Israel_, Jul 10 2017
%t A143549 nmax = 21; sol = {a[0] -> 1};
%t A143549 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*A[x]^4*A[-x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A143549 sol /. Rule -> Set;
%t A143549 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o A143549 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,2*n,A=1+x*A^4*subst(A^1,x,-x));polcoeff(A,n)}
%Y A143549 Cf. A143552, A143553, A143554.
%Y A143549 Cf. A143550, A143547.
%Y A143549 Cf. A047749, A143338, A143551.
%K A143549 nonn
%O A143549 0,3
%A A143549 _Paul D. Hanna_, Aug 24 2008