A138913
G.f. A(x) satisfies: 4*A(x) = A(A(A(x))) + 3*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 6, 99, 2362, 70484, 2463460, 97309959, 4251047468, 202470323828, 10409697289888, 573563068625768, 33682595044746416, 2099111839596600644, 138339363094940014088, 9612941947359915802978, 702527738704990333954432
Offset: 1
G.f.: A(x) = x + x^2 + 6*x^3 + 99*x^4 + 2362*x^5 + 70484*x^6 +...
A(A(x)) = x + 2*x^2 + 14*x^3 + 229*x^4 + 5456*x^5 + 162710*x^6 +...
A(A(A(x))) = x + 3*x^2 + 24*x^3 + 396*x^4 + 9448*x^5 + 281936*x^6 +...
so that 4*A(x) = A(A(A(x))) + 3*x + x^2.
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{a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n+1,A=A+polcoeff(subst(A,x,subst(A,x,A+x*O(x^i))),i)*x^i); polcoeff(A,n))}
A138915
G.f. A(x) satisfies: 6*A(x) = A(A(A(A(A(x))))) + 5*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 20, 1070, 82620, 7950630, 893138136, 113042205894, 15776443441194, 2393774318253534, 391021817774684352, 68276246115093735882, 12675272091572931300360, 2491402163326687657447940
Offset: 1
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.
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{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))}
A138916
G.f. A(x) satisfies: 7*A(x) = A(A(A(A(A(A(x)))))) + 6*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 30, 2385, 273560, 39078970, 6512700536, 1222156339336, 252751878117712, 56798072762849412, 13733835430565197700, 3548014267149570778764, 974073193845291808779496, 283008950620416071533339000
Offset: 1
G.f.: A(x) = x + x^2 + 30*x^3 + 2385*x^4 + 273560*x^5 +...
A(A(x)) = x + 2*x^2 + 62*x^3 + 4921*x^4 + 564280*x^5 +...
A(A(A(x))) = x + 3*x^2 + 96*x^3 + 7614*x^4 + 872950*x^5 +...
A(A(A(A(x)))) = x + 4*x^2 + 132*x^3 + 10470*x^4 + 1200384*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 170*x^3 + 13495*x^4 + 1547420*x^5 +...
A(A(A(A(A(A(x)))))) = x + 6*x^2 + 210*x^3 + 16695*x^4 + 1914920*x^5 +...
so that 7*A(x) = A(A(A(A(A(A(x)))))) + 6*x + x^2.
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{a(n)=local(A=x+x^2,G);if(n<1,0,for(i=3,n+1,G=x; for(j=1,6,G=subst(A,x,G+x*O(x^i)));A=A+polcoeff(G,i)*x^i);polcoeff(A,n))}
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