A112107 -id:A112107 - cp's OEIS frontend

A112106 Unique sequence of numbers {1,2,3} where g.f. A(x) satisfies A(x) = B(B(B(x))) (3rd self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

1, 3, 3, 3, 2, 2, 1, 2, 1, 3, 1, 1, 3, 3, 3, 2, 3, 3, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 1, 3, 2, 1, 3, 2, 2, 1, 2, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 2, 3, 3, 3, 3, 3, 3, 1, 1, 2, 2, 3, 3, 1, 3, 2, 1, 2, 2, 1, 1, 3, 1

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: A(x) = x + 3*x^2 + 3*x^3 + 3*x^4 + 2*x^5 + 2*x^6 + ...
then A(x) = B(B(B(x))) where
B(x) = x + x^2 - x^3 + 3*x^4 - 10*x^5 + 35*x^6 - 119*x^7 + ...
is the g.f. of A112107.

Cf. A112107, A112104, A112105, A112108-A112127.

{a(n,m=3)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

A112108 Unique sequence of numbers {1,2,3,4} where g.f. A(x) satisfies A(x) = B(B(B(B(x)))) (4th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

Original entry on oeis.org

1, 4, 4, 2, 4, 2, 4, 4, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 3, 4, 3, 2, 4, 1, 2, 4, 2, 3, 1, 4, 2, 4, 3, 1, 4, 4, 4, 2, 2, 2, 3, 3, 2, 3, 2, 2, 4, 1, 4, 2, 2, 1, 4, 3, 3, 3, 1, 1, 3, 3, 4, 4, 3, 3, 3, 3, 1, 4, 4, 3, 2, 4, 2, 2, 2, 1, 3, 4, 2, 3, 3, 1, 4, 2, 3, 1, 1, 3, 3, 4, 2, 4, 3, 1, 4, 3, 2, 1, 1, 1, 2, 1, 4, 4

Offset: 1

Paul D. Hanna, Aug 27 2005

nonn

			G.f.: A(x) = x + 4*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + ...
then A(x) = B(B(B(B(x)))) where
B(x) = x + x^2 - 2*x^3 + 8*x^4 - 38*x^5 + 194*x^6 - 992*x^7 + ...
is the g.f. of A112109.

Cf. A112109, A112104-A112107, A112110-A112127.

{a(n,m=4)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); G=F+x*O(x^n);for(i=1,m-1,G=subst(F,x,G)); return(polcoeff(G,n,x)))}

A112109 G.f. A(x) satisfies A(A(A(A(x)))) = B(x) (4th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4}, with B(0) = 0.

Original entry on oeis.org

1, 1, -2, 8, -38, 194, -992, 4777, -19831, 56116, 48008, -2062286, 16053636, -70193968, 155216743, -968038798, 23817048561, -233579083166, 333773365, 21684104628935, -121906541882294, -2171063003748135, 30425707365005935, 192144123118329872

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - 2*x^3 + 8*x^4 - 38*x^5 + 194*x^6 - 992*x^7 +...
where A(A(A(A(x)))) =
x + 4*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 4*x^8 +...
is the g.f. of A112108.

Cf. A112108, A112104-A112107, A112110-A112127.

{a(n,m=4)=local(F=x+x^2+x*O(x^n),G);if(n<1,0, for(k=3,n, G=F+x*O(x^k);for(i=1,m-1,G=subst(F,x,G)); F=F-((polcoeff(G,k)-1)\m)*x^k); return(polcoeff(F,n,x)))}

cp's OEIS Frontend

A112106 Unique sequence of numbers {1,2,3} where g.f. A(x) satisfies A(x) = B(B(B(x))) (3rd self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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A112108 Unique sequence of numbers {1,2,3,4} where g.f. A(x) satisfies A(x) = B(B(B(B(x)))) (4th self-COMPOSE) such that B(x) is an integer series, with A(0) = 0.

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A112109 G.f. A(x) satisfies A(A(A(A(x)))) = B(x) (4th self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3,4}, with B(0) = 0.

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