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A367385 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)).

%I A367385 #18 Dec 24 2023 16:06:32
%S A367385 1,1,5,61,1329,43841,1987153,116322249,8430315169,733890562273,
%T A367385 75025552012641,8851196086238969,1188516164483406289,
%U A367385 179619377095898214801,30271231938826215582001,5645050489627807288153321,1157185379272549414363693377,259281400277115714365664526529
%N A367385 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)).
%H A367385 Paul D. Hanna, <a href="/A367385/b367385.txt">Table of n, a(n) for n = 0..200</a>
%F A367385 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A367385 (1) A(x/A(x)) = exp(x*A(x)).
%F A367385 (2) A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).
%F A367385 (3) A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) = x / Series_Reversion(x*A(x)).
%e A367385 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1329*x^4/4! + 43841*x^5/5! + 1987153*x^6/6! + 116322249*x^7/7! + 8430315169*x^8/8! + 733890562273*x^9/9! + ...
%e A367385 where A(x/A(x)) = exp(x*A(x)) and
%e A367385 exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 329*x^4/4! + 8396*x^5/5! + 318577*x^6/6! + 16388086*x^7/7! + 1075939601*x^8/8! + 86549687704*x^9/9! + ...
%e A367385 Also,
%e A367385 A(x) = exp(x*B(x)^2) where B(x) = A(x*B(x)) begins
%e A367385 B(x) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2989*x^4/4! + 115136*x^5/5! + 5899159*x^6/6! + 381657928*x^7/7! + 30082660633*x^8/8! + 2814548348224*x^9/9! + ...
%e A367385 B(x)^2 = 1 + 2*x + 16*x^2/2! + 266*x^3/3! + 7168*x^4/4! + 275842*x^5/5! + 14058520*x^6/6! + 903187826*x^7/7! + 70653972896*x^8/8! + 6560662418306*x^9/9! + ...
%e A367385 Further,
%e A367385 A(x/C(x)^2) = exp(x) where C(x) = A(x/C(x)) begins
%e A367385 C(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 533*x^4/4! + 16096*x^5/5! + 680827*x^6/6! + 37544368*x^7/7! + 2577391273*x^8/8! + 213306280480*x^9/9! + ...
%e A367385 C(x)^2 = 1 + 2*x + 8*x^2/2! + 74*x^3/3! + 1344*x^4/4! + 39202*x^5/5! + 1618456*x^6/6! + 87693090*x^7/7! + 5940234656*x^8/8! + 486479747906*x^9/9! + ...
%o A367385 (PARI) {a(n) = my(A=1+x); for(i=0,n, A = exp( (1/x)*serreverse( x/(A + x*O(x^n)) )^2 )); n!*polcoeff(A,n)}
%o A367385 for(n=0,20, print1(a(n),", "))
%Y A367385 Cf. A144681.
%K A367385 nonn
%O A367385 0,3
%A A367385 _Paul D. Hanna_, Dec 22 2023