A112108 -id:A112108

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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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..105.

EXAMPLE

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.

PROG

(PARI) {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)))}

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 27 2005

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..27.

EXAMPLE

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.

PROG

(PARI) {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)))}

CROSSREFS

Cf. A112106, A112104, A112105, A112108-A112127.

KEYWORD

sign

AUTHOR

Paul D. Hanna, Aug 27 2005

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..24.

EXAMPLE

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.

PROG

(PARI) {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)))}

CROSSREFS

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

KEYWORD

sign

AUTHOR

Paul D. Hanna, Aug 27 2005

STATUS

approved

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