OFFSET
1,2
COMMENTS
Radius of convergence is r = (sqrt(85) - 9)/2, where r = r^3/(1-9*r), with A(r) = 1.
Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).
FORMULA
From Paul D. Hanna, Mar 17 2024: (Start)
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1 - 9*x) ).
(2) A( x/(1 + 9*x) )^3 = A( x^3/(1 + 9*x)^2 ).
(3) A( x/(1 + 3*x + 9*x^2) )^3 = A( x^3/(1 - 27*x^3)^2 ). (End)
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 957*x^5 + 7497*x^6 + 60233*x^7 + 492558*x^8 + 4080897*x^9 + 34152449*x^10 + 288107376*x^11 + ...
where A(x)^3 = A( x^3/(1-3*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 732*x^6 + 6615*x^7 + 59778*x^8 + 540207*x^9 + 4881870*x^10 + 44118351*x^11 + 398712097*x^12 + 3603351699*x^13 + ...
(A(x)/x)^(1/3) = 1 + x + 5*x^2 + 32*x^3 + 225*x^4 + 1672*x^5 + 12873*x^6 + 101574*x^7 + 816050*x^8 + 6647378*x^9 + 54742914*x^10 + 454832564*x^11 + ...
A( x/(1 + 3*x + 9*x^2) ) = x + 19*x^4 + 482*x^7 + 13946*x^10 + 444438*x^13 + 15330112*x^16 + 564221847*x^19 + 21863841462*x^22 + 881431824107*x^25 + 36605787985301*x^28 + 1554163122195738*x^31 + 67078838997215060*x^34 + 2931316135685487004*x^37 + ...
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 3*x + 9*x^2 + 19*x^3 - 171*x^5 - 601*x^6 + 8658*x^8 + 34409*x^9 - 576954*x^11 - 2416249*x^12 + 43795764*x^14 + 188941890*x^15 + ...
Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + 3*x + C2(x):
C0(x) = 1 + 19*x^3 - 601*x^6 + 34409*x^9 - 2416249*x^12 + 188941890*x^15 - 15788781918*x^18 + ...
C2(x) = 9*x^2 - 171*x^5 + 8658*x^8 - 576954*x^11 + 43795764*x^14 - 3590437581*x^17 + 309719962683*x^20 + ...
then C0(x) = 9*x^2/C2(x).
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-9*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 08 2015
STATUS
approved