A213009
G.f. A(x) satisfies: A(A(x)) = x+x^2 + x*A(A(A(A(x)))).
Original entry on oeis.org
1, 1, 1, 5, 21, 125, 825, 6133, 49925, 439417, 4142945, 41544161, 440710117, 4924691541, 57766255689, 709205703565, 9090541134373, 121389729560633, 1685431945085489, 24289856880005441, 362776874949660485, 5606980244843123077, 89560387072919814553
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 5*x^4 + 21*x^5 + 125*x^6 + 825*x^7 +...
where
A(A(x)) = x + 2*x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 480*x^6 + 3296*x^7 +...
A(A(A(A(x)))) = x + 4*x^2 + 16*x^3 + 80*x^4 + 480*x^5 + 3296*x^6 +...
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{a(n)=local(A=x+x^2,B=x+2*x^2);for(i=1,n,B=x+x^2+x*subst(B,x,B+x*O(x^n)));
for(i=1,n,A=(A+subst(B,x,serreverse(A+x*O(x^n))))/2);polcoeff(A,n)}
for(n=1,31,print1(a(n),", "))
A215115
G.f. A(x) satisfies: A(A(A(x))) = G(x) where G(x) = x + 2*x^2 + x*G(G(G(x))) is the g.f. of A215114.
Original entry on oeis.org
1, 1, 1, 19, 163, 2269, 34093, 584767, 10989271, 224143489, 4910384809, 114714875755, 2841991084747, 74337591206629, 2045557726962949, 59036247882081847, 1782385894711138303, 56166016733387381449, 1843556640469175481985, 62915735570546535121891
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 19*x^4 + 163*x^5 + 2269*x^6 + 34093*x^7 +...
Let G(x) = A(A(A(x))):
G(x) = x + 3*x^2 + 9*x^3 + 81*x^4 + 891*x^5 + 11907*x^6 + 184437*x^7 +...
such that G(x) = x + 2*x^2 + x*G(G(G(x))):
G(G(G(x))) = x + 9*x^2 + 81*x^3 + 891*x^4 + 11907*x^5 + 184437*x^6 +...
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{a(n)=local(A=x+x^2, B=x+2*x^2); for(i=1, n+1, B=x+2*x^2+x*subst(B, x, subst(B, x, B+x*O(x^n))));
for(j=1, n+1, A=round((2*A+subst(B, x, serreverse(subst(A,x,A+x*O(x^n)))))/3));; polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))
A215117
G.f. A(x) satisfies: A(A(A(A(x)))) = G(x) where G(x) = x + 3*x^2 + x*G(G(G(G(x)))) is the g.f. of A215116.
Original entry on oeis.org
1, 1, 1, 49, 721, 17281, 452065, 13511953, 443435185, 15816390241, 606861668161, 24867738772849, 1082158542264721, 49785517156216897, 2412544311495241633, 122762020478952148177, 6542028190536528941425, 364254737003651267997985, 21146448814786605196994305
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 49*x^4 + 721*x^5 + 17281*x^6 + 452065*x^7 +...
Let G(x) = A(A(A(A(x)))):
G(x) = x + 4*x^2 + 16*x^3 + 256*x^4 + 4864*x^5 + 111616*x^6 + 2983936*x^7 +...
such that G(x) = x + 3*x^2 + x*G(G(G(G(x)))):
G(G(G(G(x)))) = x + 16*x^2 + 256*x^3 + 4864*x^4 + 111616*x^5 + 2983936*x^6 +...
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{a(n)=local(A=x+x^2,B=x+4*x^2);for(i=1,n+1,B=x+3*x^2+x*subst(B,x,subst(B,x,subst(B,x,B+x^2*O(x^n)))));
for(j=1, n+1, A=round((3*A+subst(B, x, serreverse(subst(A,x,subst(A,x,A+x^2*O(x^n))))))/4));; polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))
A215118
G.f. satisfies: A(x) = x + 4*x^2 + x*A(A(A(A(A(x))))).
Original entry on oeis.org
1, 5, 25, 625, 18125, 628125, 25390625, 1158515625, 58308203125, 3190470703125, 187941103515625, 11832996337890625, 791834056298828125, 56063448811767578125, 4184231129351806640625, 328154000925299072265625, 26970505516268341064453125, 2317475342690856231689453125
Offset: 1
G.f.: A(x) = x + 5*x^2 + 25*x^3 + 625*x^4 + 18125*x^5 + 628125*x^6 + ...
where
A(A(A(A(x)))) = x + 25*x^2 + 625*x^3 + 18125*x^4 + 628125*x^5 + ...
Related expansions.
Let E(E(E(E(E(x))))) = A(x), then E(x) is an integer series where:
E(x) = x + x^2 + x^3 + 101*x^4 + 2301*x^5 + 82601*x^6 + 3287001*x^7 + ...
where the coefficients of E(x) are congruent to 1 modulo 100.
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{a(n) = my(A=x+4*x^2); for(i=1,n,A=x+4*x^2+x*subst(A,x,subst(A,x,subst(A,x,subst(A,x,A+x*O(x^n)))))); polcoef(A, n)}
for(n=1, 31, print1(a(n), ", "))
Showing 1-4 of 4 results.
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