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G.f. A(x) satisfies A( (x + 3*A(x)^2)^3 ) = A(x)^3.
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%I #14 Oct 14 2024 05:37:26

%S 1,3,18,136,1152,10458,99473,978480,9872181,101598389,1062382809,

%T 11255336235,120555453344,1303305334704,14202627395202,

%U 155847144409224,1720542786453765,19096869133735155,212977164179543266,2385405242723601582,26820428322385799784,302611771988083401990

%N G.f. A(x) satisfies A( (x + 3*A(x)^2)^3 ) = A(x)^3.

%H Paul D. Hanna, <a href="/A376224/b376224.txt">Table of n, a(n) for n = 1..500</a>

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

%F (1) A(x)^3 = A( (x + 3*A(x)^2)^3 ).

%F (2) x = A( x*(1 - x*G(x))^3 ), where G(x) is the g.f. of A352702.

%F (3) x = A( x - 3*x^2 - x^4*G(x^3) ), where G(x) is the g.f. of A352702.

%F a(n) ~ c * d^n / n^(3/2), where d = 12.108643088449238597222614925208058784697264797459219306522454237465345359... and c = 0.0455800108980650629231383349217685291247499776153219609599892816651... - _Vaclav Kotesovec_, Oct 14 2024

%e 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 + ...

%e where A( (x + 3*A(x)^2)^3 ) = A(x)^3.

%e RELATED SERIES.

%e 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 + ...

%e ( 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 + ...

%e Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then

%e 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) + ...

%e 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:

%e 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 + ...

%o (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)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A352702, A271960.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 13 2024