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

1, 3, 18, 127, 966, 7686, 63068, 529503, 4526262, 39262658, 344789172, 3059733222, 27400769364, 247345475628, 2248572742200, 20570124766951, 189238723449318, 1749776993081730, 16253403563598516, 151604206816149210, 1419457992097097340, 13336331712054463644, 125697697304515725840

Offset: 1

Author

Paul D. Hanna, Nov 08 2024

Keywords

• nonn

Comments

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).

Conjecture: a(n) is odd iff n = 2^k for some k >= 0.

Examples

			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 + ...
where A(x)^3 = A( x^3 + 9*x*A(x)^3 ).
RELATED SERIES.
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 + ...
Series reversion of A(x) equals B(x) - 3*x^3/B(x) where
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 + ...
so that A( B(x) - 3*x^3/B(x) ) = x.
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.0913017665091460949496315519875858022728583060252844...
  where 1/216 = A( t^3 + t/24 ).
A(t) = 1/9 at t = 0.0756231400530157002966336216229658355706050775929719...
  where 1/729 = A( t^3 + t/81 ).
A(1/11) = 0.16461186433566159924255427988603576152486558542514...
A(1/12) = 0.13356888809515041673070959997705841146178687774042...
A(1/13) = 0.11450357672473104104332015691591377007745191359804...
A(1/15) = 0.09064971528132540512370615784788517775098854995359...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Crossrefs

Cf. A271934.

Programs

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)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^3 = A( x^3 + 9*x*A(x)^3 ).

(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 ).