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A372522 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = Sum_{k>=1} k * 18^(k-1) * x^k.

%I A372522 #25 May 06 2024 10:12:47
%S A372522 0,1,6,-18,378,-5670,52488,930204,-55108026,575622774,46483766460,
%T A372522 -1494416264796,-85327731650772,5947844644410876,192190798316367540,
%U A372522 -29067440301002581416,-418574641900663175706,179341053539746099078422
%N A372522 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = Sum_{k>=1} k * 18^(k-1) * x^k.
%H A372522 Seiichi Manyama, <a href="/A372522/b372522.txt">Table of n, a(n) for n = 0..200</a>
%F A372522 Define the sequence b(n,m) as follows. If n<m, b(n,m) = 0, else if n=m, b(n,m) = 1, otherwise b(n,m) = 1/6 * ( 18^(n-m) * binomial(n+m-1,2*m-1) - Sum_{l=m+1..n-1} (b(n,l) + Sum_{k=l..n} (b(n,k) + Sum_{j=k..n} (b(n,j) + Sum_{i=j..n} (b(n,i) + Sum_{h=i..n} b(n,h) * b(h,i)) * b(i,j)) * b(j,k)) * b(k,l)) * b(l,m) ). a(n) = b(n,1).
%F A372522 Let F(x) = x/(1 - 18*x)^2, B(x) = A(A(x)) and C(x) = A(A(A(x))).
%F A372522 B(B(B(x))) = C(C(x)) = F(x).
%F A372522 B(x) = G(2*x)/2, where G(x) is the g.f. for A372499.
%F A372522 C(x) = H(9*x)/9, where H(x) is the g.f. for A309509.
%e A372522 A(A(x)) = x + 12*x^2 + 36*x^3 + 432*x^4 - 62208*x^6 + 2846016*x^7 - ...
%e A372522 A(A(A(x))) = x + 18*x^2 + 162*x^3 + 1458*x^4 + 13122*x^5 + 2125764*x^7 + ...
%Y A372522 Cf. A309509, A372492, A372499, A372520.
%K A372522 sign
%O A372522 0,3
%A A372522 _Seiichi Manyama_, May 04 2024