%I #12 Jan 04 2024 09:00:45

%S 1,1,7,136,4933,275536,21309139,2137447936,266227499017,

%T 39924910381312,7045914488563711,1437809941831499776,

%U 334581893955246072205,87792555944973238718464,25735892905876612366925515,8363132129019712402301648896,2992768723058093966270081891089

%N Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)).

%C Conjecture: a(n) == 1 (mod 3) for n >= 0.

%C Conjecture: a(2*n) == 1 (mod 2) for n >= 0.

%C Conjecture: a(2*n+1) == 0 (mod 2) for n >= 1.

%H Paul D. Hanna, <a href="/A368630/b368630.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

%F (1) A(x/A(x)^2) = exp(x*A(x)).

%F (2) A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) = ( (1/x)*Series_Reversion(x/A(x)^2) )^(1/2).

%F (3) A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) = x/Series_Reversion(x*A(x)).

%e E.g.f.: A(x) = 1 + x + 7*x^2/2! + 136*x^3/3! + 4933*x^4/4! + 275536*x^5/5! + 21309139*x^6/6! + 2137447936*x^7/7! + 266227499017*x^8/8! + ...

%e where A(x/A(x)^2) = exp(x*A(x)) and

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 28*x^3/3! + 653*x^4/4! + 28096*x^5/5! + 1833367*x^6/6! + 162874048*x^7/7! + ...

%e Also,

%e A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) begins

%e B(x) = 1 + x + 11*x^2/2! + 292*x^3/3! + 13149*x^4/4! + 861376*x^5/5! + 75412591*x^6/6! + 8365301568*x^7/7! + ...

%e B(x)^2 = 1 + 2*x + 24*x^2/2! + 650*x^3/3! + 29360*x^4/4! + 1918482*x^5/5! + 167206144*x^6/6! + ...

%e B(x)^3 = 1 + 3*x + 39*x^2/2! + 1080*x^3/3! + 49029*x^4/4! + 3199728*x^5/5! + 277840179*x^6/6! + ...

%e Further,

%e A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) begins

%e C(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 2889*x^4/4! + 154441*x^5/5! + 11527693*x^6/6! + 1120674717*x^7/7! + ...

%e C(x)^3 = 1 + 3*x + 21*x^2/2! + 351*x^3/3! + 11337*x^4/4! + 582843*x^5/5! + 42300765*x^6/6! + ...

%o (PARI) {a(n) = my(A=1+x); for(i=0,n, A = exp( x*((1/x)*serreverse( x/(A^2 + x*O(x^n)) ))^(3/2) )); n!*polcoeff(A,n)}

%o for(n=0,20, print1(a(n),", "))

%Y Cf. A368631, A368632, A144682, A144683, A367385.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 02 2024