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(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.
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allocated for Paul D. Hanna
Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)^2).
1, 1, 9, 217, 9521, 634321, 58026745, 6846238057, 998806698209, 174849870369313, 35915074166268521, 8507730512772340345, 2292605150744212481809, 695028316821630097748209, 234883073320203308189545049, 87808334177056337272289692681, 36075481332626610937457504918465
0,3
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x/A(x)^2) = exp(x*A(x)^2).
(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).
(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).
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! + ...
where A(x/A(x)^2) = exp(x*A(x)^2) and
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! + ...
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! + ...
Also,
A(x) = exp(x*B(x)^4) where B(x) = A(x*B(x)^2) begins
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! + ...
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! + ...
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! + ...
Further,
A(x/C(x)^4) = exp(x) where C(x) = A(x/C(x)^2) begins
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! + ...
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! + ...
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! + ...
(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)}
for(n=0, 20, print1(a(n), ", "))
Cf. A367385.
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Paul D. Hanna, Jan 01 2024
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