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A368632 Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)^2).

%I A368632 #9 Jan 04 2024 08:32:29
%S A368632 1,1,9,217,9521,634321,58026745,6846238057,998806698209,
%T A368632 174849870369313,35915074166268521,8507730512772340345,
%U A368632 2292605150744212481809,695028316821630097748209,234883073320203308189545049,87808334177056337272289692681,36075481332626610937457504918465
%N A368632 Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)^2).
%F A368632 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A368632 (1) A(x/A(x)^2) = exp(x*A(x)^2).
%F A368632 (2) A(x) = exp(x*B(x)^4) where B(x) = A(x*B(x)^2) = ( (1/x)*Series_Reversion(x/A(x)^2) )^(1/2).
%F A368632 (3) A(x/C(x)^4) = exp(x) where C(x) = A(x/C(x)^2) = ( x/Series_Reversion(x*A(x)^2) )^(1/2).
%F A368632 (4) A(x)^2 = F(2*x) where F(x/F(x)) = exp(x*F(x)) and F(x) is the e.g.f. of A367385.
%e A368632 E.g.f.: A(x) = 1 + x + 9*x^2/2! + 217*x^3/3! + 9521*x^4/4! + 634321*x^5/5! + 58026745*x^6/6! + 6846238057*x^7/7! + 998806698209*x^8/8! + ...
%e A368632 where A(x/A(x)^2) = exp(x*A(x)^2) and
%e A368632 exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2265*x^4/4! + 119361*x^5/5! + 9255133*x^6/6! + 965731593*x^7/7! + ...
%e A368632 A(x)^2 = 1 + 2*x + 20*x^2/2! + 488*x^3/3! + 21264*x^4/4! + 1402912*x^5/5! + 127177792*x^6/6! + 14889247872*x^7/7! + ...
%e A368632 Also,
%e A368632 A(x) = exp(x*B(x)^4) where B(x) = A(x*B(x)^2) begins
%e A368632 B(x) = 1 + x + 13*x^2/2! + 409*x^3/3! + 21769*x^4/4! + 1680161*x^5/5! + 172774357*x^6/6! + 22446379705*x^7/7! + ...
%e A368632 B(x)^2 = 1 + 2*x + 28*x^2/2! + 896*x^3/3! + 47824*x^4/4! + 3684352*x^5/5! + 377546176*x^6/6! + ...
%e A368632 B(x)^4 = 1 + 4*x + 64*x^2/2! + 2128*x^3/3! + 114688*x^4/4! + 8826944*x^5/5! + 899745280*x^6/6! + ...
%e A368632 Further,
%e A368632 A(x/C(x)^4) = exp(x) where C(x) = A(x/C(x)^2) begins
%e A368632 C(x) = 1 + x + 5*x^2/2! + 97*x^3/3! + 3801*x^4/4! + 233681*x^5/5! + 20005213*x^6/6! + 2225362161*x^7/7! + ...
%e A368632 C(x)^2 = 1 + 2*x + 12*x^2/2! + 224*x^3/3! + 8528*x^4/4! + 515072*x^5/5! + 43572928*x^6/6! + ...
%e A368632 C(x)^4 = 1 + 4*x + 32*x^2/2! + 592*x^3/3! + 21504*x^4/4! + 1254464*x^5/5! + 103581184*x^6/6! + ...
%o A368632 (PARI) {a(n) = my(A=1+x); for(i=0,n, A = exp( x*((1/x)*serreverse( x/(A^2 + x*O(x^n)) ))^2 )); n!*polcoeff(A,n)}
%o A368632 for(n=0,20, print1(a(n),", "))
%Y A368632 Cf. A144682, A367385.
%K A368632 nonn
%O A368632 0,3
%A A368632 _Paul D. Hanna_, Jan 01 2024