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A141121 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 36*x^2.

%I A141121 #16 May 05 2024 08:54:39
%S A141121 1,6,-180,8640,-498960,31434480,-2055943296,135216506304,
%T A141121 -8720972739072,538646016002688,-31024094144060160,
%U A141121 1609593032459782656,-71392972690228672512,2461961564459510280192,-51302015299696881770496,-415041229811424576835584
%N A141121 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 36*x^2.
%H A141121 Seiichi Manyama, <a href="/A141121/b141121.txt">Table of n, a(n) for n = 1..200</a>
%F A141121 From _Seiichi Manyama_, May 04 2024: (Start)
%F A141121 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 * ( 36^(n-m) * binomial(m,n-m) - 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 A141121 A(A(x)) = F(4*x)/4, where F(x) is the g.f. for A141118.
%F A141121 A(A(A(x))) = G(9*x)/9, where G(x) is the g.f. for A027436. (End)
%e A141121 G.f.: A(x) = x + 6*x^2 - 180*x^3 + 8640*x^4 - 498960*x^5 +...
%e A141121 A(A(x)) = x + 12*x^2 - 288*x^3 + 12096*x^4 - 622080*x^5 +...
%e A141121 A(A(A(x))) = x + 18*x^2 - 324*x^3 + 11664*x^4 - 524880*x^5 +...
%e A141121 A(A(A(A(x)))) = x + 24*x^2 - 288*x^3 + 8640*x^4 - 331776*x^5 +...
%e A141121 A(A(A(A(A(x))))) = x + 30*x^2 - 180*x^3 + 4320*x^4 - 136080*x^5 +...
%o A141121 (PARI) {a(n, m=6)=local(F=x+m*x^2+x*O(x^n), G); if(n<1, 0, for(k=3, n, G=F+x*O(x^k); for(i=1, m-1, G=subst(F, x, G)); F=F+((-polcoeff(G, k))/m)*x^k); return(polcoeff(F, n, x)))}
%Y A141121 Cf. A027436, A141118, A141119, A141120.
%K A141121 sign
%O A141121 1,2
%A A141121 _Paul D. Hanna_, Jun 05 2008