OFFSET
1,3
COMMENTS
Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
First negative term is a(83). Georg Fischer, Feb 15 2019
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
G.f. satisfies:
(1) A(x - A(x^3 - x^7)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1)/x / (n+1)! ).
EXAMPLE
G.f.: A(x) = x + x^3 + 3*x^5 + 11*x^7 + 46*x^9 + 207*x^11 + 977*x^13 + 4767*x^15 + 23835*x^17 + 121424*x^19 + 627747*x^21 + 3284055*x^23 + 17348254*x^25 +...
such that A(x) = x + A( A(x)^3 - A(x)^7 ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 52*x^9 + 240*x^11 + 1155*x^13 + 5727*x^15 + 29034*x^17 + 149727*x^19 + 782627*x^21 + 4135668*x^23 + 22051158*x^25 +...
A(x)^7 = x^7 + 7*x^9 + 42*x^11 + 238*x^13 + 1323*x^15 + 7308*x^17 + 40327*x^19 + 222804*x^21 + 1233624*x^23 + 6847281*x^25 + 38102099*x^27 +...
A(x^3 - x^7) = x^3 - x^7 + x^9 - 3*x^13 + 3*x^15 + 3*x^17 - 15*x^19 + 10*x^21 + 30*x^23 - 77*x^25 + 16*x^27 + 231*x^29 - 399*x^31 - 178*x^33 + 1653*x^35 - 1892*x^37 - 2887*x^39 +...
where Series_Reversion(A(x)) = x - A(x^3 - x^7).
PROG
(PARI) {a(n) = my(A=x); for(i=1, 2*n, A = x + subst(A, x, A^3 - A^7 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Aug 20 2016
STATUS
approved