A112105 G.f. A(x) satisfies A(A(x)) = B(x) such that the coefficients of B(x) consist of all 1's and 2's, with A(0) = 0.

1, 1, 0, 0, 1, -3, 7, -10, -5, 84, -248, 90, 2160, -7541, -5846, 122824, -186259, -2036532, 8665409, 36714511, -345711246, -517802065, 14415153844, -9423161197, -653074772917, 1896978939457, 32374651932128, -184814895196023, -1733326272860598, 16839263882542991, 96742403684106435

Offset

1,6

Links

Kenny Lau, Table of n, a(n) for n = 1..553

Example

A(x) = x + x^2 + x^5 - 3*x^6 + 7*x^7 - 10*x^8 - 5*x^9 +...
where A(A(x)) = x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 +... is the g.f. of A112104.

Mathematica

kmax = 40;
A[x_] := Sum[a[k] x^k, {k, kmax}];
B[x_] := Sum[b[k] x^k, {k, kmax}];
sol = {a[1] -> 1, b[1] -> 1};
Do[sc = SeriesCoefficient[A[(A[x] /. sol) + O[x]^(k+1)] - B[x], {x, 0, k}] /. sol; r = Reduce[(b[k] == 1 || b[k] == 2) && sc == 0, {a[k], b[k]}, Integers]; sol = Join[r // ToRules, sol], {k, 2, kmax}];
a /@ Range[kmax] /. sol (* Jean-François Alcover, Nov 05 2019 *)

Prog

(PARI) {a(n, m=2)=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. A112104, A112106-A112127.

Sequence in context: A118559 A189244 A127789 * A065501 A144385 A144399

Adjacent sequences: A112102 A112103 A112104 * A112106 A112107 A112108

Keyword

sign

Author

Paul D. Hanna, Aug 27 2005

Status

approved