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),", "))
A215116
G.f. satisfies: A(x) = x + 3*x^2 + x*A(A(A(A(x)))).
Original entry on oeis.org
1, 4, 16, 256, 4864, 111616, 2983936, 89743360, 2970861568, 106768629760, 4125849419776, 170207219286016, 7454572671926272, 345078981839552512, 16822127738969128960, 860944587541763325952, 46137178395559050870784, 2582843669636660403896320, 150735442996358913332346880
Offset: 1
G.f.: A(x) = x + 4*x^2 + 16*x^3 + 256*x^4 + 4864*x^5 + 111616*x^6 + 2983936*x^7 +...
where
A(A(A(A(x)))) = x + 16*x^2 + 256*x^3 + 4864*x^4 + 111616*x^5 + 2983936*x^6 +...
Related expansions.
Let D(D(D(D(x)))) = A(x), then D(x) is an integer series where:
D(x) = x + x^2 + x^3 + 49*x^4 + 721*x^5 + 17281*x^6 + 452065*x^7 +...
where the coefficients of D(x) are congruent to 1 modulo 48.
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{a(n)=local(A=x+4*x^2); for(i=1,n,A=x+3*x^2+x*subst(A,x,subst(A,x,subst(A,x,A+x*O(x^n))))); 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), ", "))
A215119
G.f. A(x) satisfies: A(A(A(A(A(x))))) = G(x) where G(x) = x + 4*x^2 + x*G(G(G(G(G(x))))) is the g.f. of A215118.
Original entry on oeis.org
1, 1, 1, 101, 2301, 82601, 3287001, 149411501, 7474902101, 406765054801, 23836604715601, 1493376284080501, 99459838574595501, 7009748111184956601, 520845172037612209801, 40672220108202107951101, 3328819620490715884626501, 284871268231239093741932001
Offset: 1
G.f.: A(x) = x + x^2 + x^3 + 101*x^4 + 2301*x^5 + 82601*x^6 + 3287001*x^7 +...
Let G(x) = A(A(A(A(A(x))))):
G(x) = x + 5*x^2 + 25*x^3 + 625*x^4 + 18125*x^5 + 628125*x^6 + 25390625*x^7 +...
such that G(x) = x + 4*x^2 + x*G(G(G(G(G(x))))):
G(G(G(G(G(x))))) = x + 25*x^2 + 625*x^3 + 18125*x^4 + 628125*x^5 + 25390625*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+4*x^2+x*subst(B,x,subst(B,x,subst(B,x,subst(B,x,B+x^2*O(x^n))))));
for(j=1, n+1, A=round((4*A+subst(B, x, serreverse(subst(A,x,subst(A,x,subst(A,x,A+x^2*O(x^n)))))))/5));; polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))
Showing 1-4 of 4 results.
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