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A143047 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^4.

%I A143047 #17 Jul 08 2025 07:46:43
%S A143047 1,1,-4,-10,84,265,-2604,-8900,94692,337940,-3767312,-13812674,
%T A143047 158785964,593029550,-6967201736,-26372738120,314904180100,
%U A143047 1204230041900,-14560722724912,-56130528427400,685514219386576,2659770565898729,-32749512944380172
%N A143047 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^4.
%H A143047 Seiichi Manyama, <a href="/A143047/b143047.txt">Table of n, a(n) for n = 0..500</a>
%F A143047 G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^4)^4.
%F A143047 G.f. satisfies: [A(x)^5 + A(-x)^5]/2 = [A(x)^4 + A(-x)^4]/2.
%F A143047 a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - _Seiichi Manyama_, Jul 08 2025
%e A143047 A(x) = 1 + x - 4*x^2 - 10*x^3 + 84*x^4 + 265*x^5 - 2604*x^6 - 8900*x^7 +...
%e A143047 A(x)^4 = 1 + 4*x - 10*x^2 - 84*x^3 + 265*x^4 + 2604*x^5 - 8900*x^6 -...
%e A143047 A(x)^5 = 1 + 5*x - 10*x^2 - 120*x^3 + 265*x^4 + 3906*x^5 - 8900*x^6 -...
%e A143047 Note that a bisection of A^5 equals a bisection of A^4.
%o A143047 (PARI) a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^4);polcoeff(A,n)
%Y A143047 Cf. A143045, A143046, A143048, A143049, A213252, A213281, A213335.
%Y A143047 Cf. A171202.
%K A143047 sign
%O A143047 0,3
%A A143047 _Paul D. Hanna_, Jul 19 2008