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%I #17 Mar 18 2024 12:53:22
%S 1,3,18,127,957,7497,60233,492558,4080897,34152449,288107376,
%T 2446274610,20883006135,179081408925,1541668556502,13316391292552,
%U 115359341792511,1001932660939401,8722045942211055,76082885748597996,664898144584551048,5820315513644860974,51026465572312794534,447965934572491365465,3937723838880233903750
%N G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-9*x) ), with A(0) = 0.
%C Radius of convergence is r = (sqrt(85) - 9)/2, where r = r^3/(1-9*r), with A(r) = 1.
%C 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).
%F From _Paul D. Hanna_, Mar 17 2024: (Start)
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
%F (1) A(x)^3 = A( x^3/(1 - 9*x) ).
%F (2) A( x/(1 + 9*x) )^3 = A( x^3/(1 + 9*x)^2 ).
%F (3) A( x/(1 + 3*x + 9*x^2) )^3 = A( x^3/(1 - 27*x^3)^2 ). (End)
%e 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 + ...
%e where A(x)^3 = A( x^3/(1-3*x) ).
%e RELATED SERIES.
%e 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 + ...
%e (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 + ...
%e 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 + ...
%e Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
%e 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 + ...
%e Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + 3*x + C2(x):
%e C0(x) = 1 + 19*x^3 - 601*x^6 + 34409*x^9 - 2416249*x^12 + 188941890*x^15 - 15788781918*x^18 + ...
%e C2(x) = 9*x^2 - 171*x^5 + 8658*x^8 - 576954*x^11 + 43795764*x^14 - 3590437581*x^17 + 309719962683*x^20 + ...
%e then C0(x) = 9*x^2/C2(x).
%o (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)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A264228, A264229.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Nov 08 2015
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