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A143048 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^5.

%I A143048 #14 Jul 08 2025 07:46:28
%S A143048 1,1,-5,-15,165,630,-8151,-33780,474045,2052495,-30206330,-134392230,
%T A143048 2040588775,9248893360,-143569282680,-659546365020,10407737293965,
%U A143048 48303692377425,-771991701692175,-3611789245335285,58311219888996170,274581478640096340
%N A143048 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^5.
%H A143048 Seiichi Manyama, <a href="/A143048/b143048.txt">Table of n, a(n) for n = 0..500</a>
%F A143048 G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^5)^5.
%F A143048 G.f. satisfies: [A(x)^6 + A(-x)^6]/2 = [A(x)^5 + A(-x)^5]/2.
%F A143048 a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - _Seiichi Manyama_, Jul 08 2025
%e A143048 A(x) = 1 + x - 5*x^2 - 15*x^3 + 165*x^4 + 630*x^5 - 8151*x^6 -++-...
%e A143048 A(x)^5 = 1 + 5*x - 15*x^2 - 165*x^3 + 630*x^4 + 8151*x^5 - 33780*x^6 -...
%e A143048 A(x)^6 = 1 + 6*x - 15*x^2 - 220*x^3 + 630*x^4 + 11286*x^5 - 33780*x^6 -...
%e A143048 Note that a bisection of A^6 equals a bisection of A^5.
%o A143048 (PARI) a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^5);polcoeff(A,n)
%Y A143048 Cf. A143045, A143046, A143047, A143049, A213252, A213281, A213335.
%Y A143048 Cf. A171204.
%K A143048 sign
%O A143048 0,3
%A A143048 _Paul D. Hanna_, Jul 19 2008