OFFSET
3,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 3..403
FORMULA
G.f. A(x) = Sum_{n>=3} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = x^3 + x^4.
(2) A(x - x^3 - x^4 - A(x)) = (x - A(x))^3 + (x - A(x))^4.
(3) A(x) = B(x)^3 + B(x)^4 where B(x - A(x)) = x and B(x) = x + A(B(x)).
(4) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! where B(x - A(x)) = x.
(5) B(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*A(x)^n / n! ) where B(x - A(x)) = x.
EXAMPLE
A(x) = x^3 + x^4 + 3*x^5 + 7*x^6 + 25*x^7 + 78*x^8 + 303*x^9 + 1104*x^10 + 4536*x^11 + 18174*x^12 + 78042*x^13 + 333204*x^14 + ...
where A(x - A(x)) = x^3 + x^4.
RELATED SERIES.
Let B(x) satisfy B(x - A(x)) = x, then
B(x) = x + x^3 + x^4 + 6*x^5 + 14*x^6 + 65*x^7 + 213*x^8 + 943*x^9 + 3634*x^10 + 16191*x^11 + 68263*x^12 + ...
where B(x) = x + B(B(x))^3 + B(B(x))^4 and B(x) = x + A(B(x)).
Also, from B(x) given above, we have the related series
B(x)^3 = x^3 + 3*x^5 + 3*x^6 + 21*x^7 + 48*x^8 + 235*x^9 + 762*x^10 + 3432*x^11 + 13153*x^12 + ...
B(x)^4 = x^4 + 4*x^6 + 4*x^7 + 30*x^8 + 68*x^9 + 342*x^10 + 1104*x^11 + 5021*x^12 + ...
where A(x) = B(x)^3 + B(x)^4.
PROG
(PARI) \\ Using A(x) = B(x)^3 + B(x)^4 where B(x - A(x)) = x.
{a(n) = my(A = x^3); for(i=1, n, B = serreverse(x - A + x^3*O(x^n));
A = B^3 + B^4); polcoef(A, n)}
for(n=3, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 05 2026
STATUS
approved