OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( (x + 3*A(x)^2)^3 ).
(2) x = A( x*(1 - x*G(x))^3 ), where G(x) is the g.f. of A352702.
(3) x = A( x - 3*x^2 - x^4*G(x^3) ), where G(x) is the g.f. of A352702.
a(n) ~ c * d^n / n^(3/2), where d = 12.108643088449238597222614925208058784697264797459219306522454237465345359... and c = 0.0455800108980650629231383349217685291247499776153219609599892816651... - Vaclav Kotesovec, Oct 14 2024
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 136*x^4 + 1152*x^5 + 10458*x^6 + 99473*x^7 + 978480*x^8 + 9872181*x^9 + 101598389*x^10 + ...
where A( (x + 3*A(x)^2)^3 ) = A(x)^3.
RELATED SERIES.
A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 759*x^6 + 7362*x^7 + 73386*x^8 + 747567*x^9 + 7749720*x^10 + 81500094*x^11 + 867420469*x^12 + ...
( x^2*A(x) )^(1/3) = x + x^2 + 5*x^3 + 35*x^4 + 284*x^5 + 2508*x^6 + 23401*x^7 + 226950*x^8 + 2265015*x^9 + 23110418*x^10 + ...
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - 3*x^2 - x^4 - x^7 - 2*x^10 - 4*x^13 - 9*x^16 - 22*x^19 - 55*x^22 - 142*x^25 - 376*x^28 - ... + -A352702(n)*x^(3*n+4) + ...
where B(x) = x*(1 - x*G(x))^3 and B(x) = x - 3*x^2 - x^4*G(x^3), where G(x) is the g.f. of A352702 and begins:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 55*x^6 + 142*x^7 + 376*x^8 + 1011*x^9 + 2758*x^10 + ...
PROG
(PARI) {a(n) = my(A = x+x^2); for(m=1, n, A = truncate(A) + x^2*O(x^m); A = subst(A, x, (x + 3*A^2)^3 )^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 13 2024
STATUS
approved