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A377106 G.f. A(x) satisfies A(x)^3 = A( x^3 + 9*x*A(x)^3 ), with A(0)=0, A'(0)=1.

%I A377106 #18 Nov 09 2024 10:03:12
%S A377106 1,3,18,127,966,7686,63068,529503,4526262,39262658,344789172,
%T A377106 3059733222,27400769364,247345475628,2248572742200,20570124766951,
%U A377106 189238723449318,1749776993081730,16253403563598516,151604206816149210,1419457992097097340,13336331712054463644,125697697304515725840
%N A377106 G.f. A(x) satisfies A(x)^3 = A( x^3 + 9*x*A(x)^3 ), with A(0)=0, A'(0)=1.
%C A377106 Compare to C(x)^3 = C( x^3 + 3*x*C(x)^3 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C A377106 Conjecture: a(n) is odd iff n = 2^k for some k >= 0.
%H A377106 Paul D. Hanna, <a href="/A377106/b377106.txt">Table of n, a(n) for n = 1..520</a>
%F A377106 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A377106 (1) A(x)^3 = A( x^3 + 9*x*A(x)^3 ).
%F A377106 (2) A(x)^9 = A( x^9 + 27*x^7*A(x)^3 + 243*x^5*A(x)^6 + 738*x^3*A(x)^9 + 81*x*A(x)^12 ).
%e A377106 G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 966*x^5 + 7686*x^6 + 63068*x^7 + 529503*x^8 + 4526262*x^9 + 39262658*x^10 + 344789172*x^11 + 3059733222*x^12 + ...
%e A377106 where A(x)^3 = A( x^3 + 9*x*A(x)^3 ).
%e A377106 RELATED SERIES.
%e A377106 A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 732*x^6 + 6642*x^7 + 60507*x^8 + 553329*x^9 + 5079024*x^10 + 46788678*x^11 + 432520930*x^12 + ...
%e A377106 Series reversion of A(x) equals B(x) - 3*x^3/B(x) where
%e A377106 B(x) = x + 8*x^4 - 280*x^7 + 15328*x^10 - 1007576*x^13 + 73169608*x^16 - 5656895520*x^19 + 456585800584*x^22 - 38029012055320*x^25 + 3244225801946920*x^28 - 282033503420822552*x^31 + ...
%e A377106 so that A( B(x) - 3*x^3/B(x) ) = x.
%e A377106 SPECIFIC VALUES.
%e A377106 A(t) = 1/6 at t = 0.0913017665091460949496315519875858022728583060252844...
%e A377106   where 1/216 = A( t^3 + t/24 ).
%e A377106 A(t) = 1/9 at t = 0.0756231400530157002966336216229658355706050775929719...
%e A377106   where 1/729 = A( t^3 + t/81 ).
%e A377106 A(1/11) = 0.16461186433566159924255427988603576152486558542514...
%e A377106 A(1/12) = 0.13356888809515041673070959997705841146178687774042...
%e A377106 A(1/13) = 0.11450357672473104104332015691591377007745191359804...
%e A377106 A(1/15) = 0.09064971528132540512370615784788517775098854995359...
%o A377106 (PARI) {a(n) = my(A=x+3*x^2); for(m=1, n, A = truncate(A); A = subst(A, x, x^3 + 9*x*A^3 +x^4*O(x^m))^(1/3) ); polcoeff(A, n)}
%o A377106 for(n=1, 30, print1(a(n), ", "))
%Y A377106 Cf. A271934.
%K A377106 nonn
%O A377106 1,2
%A A377106 _Paul D. Hanna_, Nov 08 2024