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A381666 The generating function A(x) satisfies the functional equation: A(x)+x = x*A(A(x)).

%I A381666 #15 Mar 10 2025 16:39:50
%S A381666 0,-1,1,0,-2,1,10,-13,-70,163,585,-2162,-5361,30588,49870,-459125,
%T A381666 -411370,7257651,1513653,-119997558,56857538,2062729507,-2444340720,
%U A381666 -36662245639,71849171621,670108236318,-1904023701457,-12520858710212,48731008916451,237412587011506,-1237341547854760
%N A381666 The generating function A(x) satisfies the functional equation: A(x)+x = x*A(A(x)).
%C A381666 Shifts left under COMPOSE transform with itself.
%H A381666 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A381666 Let a(n) = b(n, 1), with b(1, m) = -1 and b(0, m) = 0, then
%F A381666 b(n, m) = Sum_{k=0..n-1} (-1)^(n-1)*m*binomial(n + m - 1, k)/(n + m - 1) * b(n - k, k).
%e A381666 G.f.: A(x) = -x + x^2 - 2*x^4 + x^5 + 10*x^6 + ...
%e A381666 A(A(x)) = x - 2*x^3 + x^4 + 10*x^5 - 13*x^6 + ...
%o A381666 (PARI) a(n) = { my(A=-1+x); for(i=0, n, A=-1+x*A*subst(A, x, x*A+x*O(x^n))); if(n==0,0,polcoeff(A, n-1))}
%Y A381666 Cf. A030266 ( A(x)-x = x*A(A(x)) ).
%Y A381666 Cf. A347080 ( A(x)-x = x*A(A(-x)) ).
%Y A381666 Cf. A381669, A381670.
%K A381666 sign,eigen
%O A381666 0,5
%A A381666 _Thomas Scheuerle_, Mar 03 2025