A112108 -id:A112108 - 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)))}

A112107 G.f. A(x) satisfies A(A(A(x))) = B(x) (3rd self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3}, with B(0) = 0.

Original entry on oeis.org

1, 1, -1, 3, -10, 35, -119, 360, -792, -33, 12779, -82525, 305861, -552011, -126321, -8385020, 138177591, -433073834, -5366414982, 51203452090, 123835509276, -4174647911014, 5274854624423, 373574363131841, -2054088594386738, -34047892948849106, 391005463740951942

Offset: 1

Paul D. Hanna, Aug 27 2005

sign

			A(x) = x + x^2 - x^3 + 3*x^4 - 10*x^5 + 35*x^6 - 119*x^7 + ...
where A(A(A(x))) = x + 3*x^2 + 3*x^3 + 3*x^4 + 2*x^5 + ...
is the g.f. of A112106.

Cf. A112106, 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); return(polcoeff(F,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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A112107 G.f. A(x) satisfies A(A(A(x))) = B(x) (3rd self-COMPOSE of A) such that the coefficients of B(x) consist only of numbers {1,2,3}, with B(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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