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G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 6*x^2 + 36*x^3.
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%I #29 May 06 2024 10:12:51

%S 0,1,1,1,-35,325,-1295,-12455,283285,-2186675,-5612255,324564625,

%T -2869315163,-12271744331,490525545169,-2159646628535,-58485623789483,

%U 634417586418781,8780962428445537,-156001827155519807,-2145519156372933275,50455156500263955781

%N G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 6*x^2 + 36*x^3.

%H Seiichi Manyama, <a href="/A371841/b371841.txt">Table of n, a(n) for n = 0..200</a>

%F 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 * ( 6^(n-m) * Sum_{l=0..m} binomial(l,n-3*m+2*l) * binomial(m,l) - 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 Let B(x) = A(A(x)) and C(x) = A(A(A(x))).

%F B(B(B(x))) = C(C(x)) = x + 6*x^2 + 36*x^3.

%F B(x) = F(2*x)/2, where F(x) is the g.f. for A220288.

%F C(x) = G(3*x)/3, where G(x) is the g.f. for A220110.

%e A(A(x)) = x + 2*x^2 + 4*x^3 - 64*x^4 + 448*x^5 - 832*x^6 - 24704*x^7 + ...

%e A(A(A(x))) = x + 3*x^2 + 9*x^3 - 81*x^4 + 405*x^5 + 243*x^6 - 28431*x^7 + ...

%Y Cf. A220110, A220288, A372521, A372537.

%K sign

%O 0,5

%A _Seiichi Manyama_, May 04 2024