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A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.

%I A361046 #28 Mar 15 2023 03:03:17
%S A361046 1,3,45,1113,36459,1448568,66726309,3469988835,200242815669,
%T A361046 12670449226269,871389659249424,64693985439491127,5156607707368927875,
%U A361046 439261264283443326927,39831856169938193953827,3831650468281643037364389,389807188331526942149375433
%N A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
%H A361046 Paul D. Hanna, <a href="/A361046/b361046.txt">Table of n, a(n) for n = 0..300</a>
%F A361046 G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
%F A361046 (1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
%F A361046 (2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^2).
%F A361046 (3) B(x - x^3*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361047.
%F A361046 (4) a(n) = (2*n+1) * A361047(n+1) for n >= 0.
%F A361046 a(n) == 0 (mod 3) for n > 0.
%F A361046 a(n) ~ c * 2^n * n! * n^alfa / LambertW(1/2)^n, where alfa = 1.623844426394406... and c = 0.18597481905555548924712403113114... - _Vaclav Kotesovec_, Mar 04 2023
%F A361046 alfa = (15*LambertW(1/2) - 1 + 3/(1 + LambertW(1/2)))/4. - _Vaclav Kotesovec_, Mar 15 2023
%e A361046 G.f.: A(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + a(n)*x^(2*n) + ...
%e A361046 where
%e A361046 A(x) = 1 + (d/dx x^3*A(x)^2) + (d^2/dx^2 x^6*A(x)^4)/2! + (d^3/dx^3 x^9*A(x)^6)/3! + (d^4/dx^4 x^12*A(x)^8)/4! + (d^5/dx^5 x^15*A(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(2*n))/n! + ...
%e A361046 Related series.
%e A361046 Let B(x) = Series_Reversion(x - x^3*A(x)^2), which begins
%e A361046 B(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + 11778989157*x^17 + ... + A361047(n)*x^(2*n-1) + ...
%e A361046 then A(x) = B'(x) and
%e A361046 B(x) = x * exp( x^2*A(x)^2 + (d/dx x^5*A(x)^4)/2! + (d^2/dx^2 x^8*A(x)^6)/3! + (d^3/dx^3 x^11*A(x)^8)/4! + (d^4/dx^4 x^14*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(2*n))/n! + ... ).
%t A361046 nt = 40; (* number of terms to produce *)
%t A361046 A[_] = 0;
%t A361046 Do[A[x_] = D[InverseSeries[x - x^3*A[x]^2 + O[x]^k] // Normal, x], {k, 1, 2*nt}];
%t A361046 CoefficientList[A[x^(1/2)], x] (* _Jean-François Alcover_, Mar 04 2023 *)
%o A361046 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A361046 {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(2*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
%o A361046 for(n=0, 20, print1(a(n), ", "))
%o A361046 (PARI) /* Using series reversion (faster) */
%o A361046 {a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^2 +O(x^(2*n+3))))); polcoeff(A, 2*n)}
%o A361046 for(n=0, 20, print1(a(n), ", "))
%Y A361046 Cf. A361047, A360950, A360973, A360974, A360975, A360976.
%Y A361046 Cf. A229619, A356848, A088714, A303063.
%K A361046 nonn
%O A361046 0,2
%A A361046 _Paul D. Hanna_, Mar 03 2023