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editing
approved
A(A(A(A(A(x))))) is the 5-th 5th self-composition of the g.f. A(x).
approved
editing
_Paul D. Hanna (pauldhanna(AT)juno.com), _, Apr 03 2008
G.f. A(x) satisfies: 6*A(x) = A(A(A(A(A(x))))) + 5*x + x^2 with A(0)=0.
1, 1, 20, 1070, 82620, 7950630, 893138136, 113042205894, 15776443441194, 2393774318253534, 391021817774684352, 68276246115093735882, 12675272091572931300360, 2491402163326687657447940
1,3
A(A(A(A(A(x))))) is the 5-th self-composition of the g.f. A(x).
G.f.: A(x) = x + x^2 + 20*x^3 + 1070*x^4 + 82620*x^5 +...
A(A(x)) = x + 2*x^2 + 42*x^3 + 2241*x^4 + 172960*x^5 +...
A(A(A(x))) = x + 3*x^2 + 66*x^3 + 3519*x^4 + 271550*x^5 +...
A(A(A(A(x)))) = x + 4*x^2 + 92*x^3 + 4910*x^4 + 378944*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 120*x^3 + 6420*x^4 + 495720*x^5 +...
so that 6*A(x) = A(A(A(A(A(x))))) + 5*x + x^2.
(PARI) {a(n)=local(A=x+x^2, G); if(n<1, 0, for(i=3, n+1, G=x; for(j=1, 5, G=subst(A, x, G+x*O(x^i))); A=A+polcoeff(G, i)*x^i); polcoeff(A, n))}
nonn
Paul D. Hanna (pauldhanna(AT)juno.com), Apr 03 2008
approved