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A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).

%I A212722 #17 Oct 08 2023 06:48:07
%S A212722 1,1,3,25,313,5341,115651,3036517,93767185,3330162073,133737097411,
%T A212722 5992748728561,296433923379529,16044427276953973,943207466055927619,
%U A212722 59848531677741706621,4076826825898115406241,296742863575079244130225
%N A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).
%H A212722 G. C. Greubel, <a href="/A212722/b212722.txt">Table of n, a(n) for n = 0..358</a>
%H A212722 Vaclav Kotesovec, <a href="http://oeis.org/A245265/a245265.pdf">Asymptotic of sequences A161630, A212722, A212917 and A245265</a>
%F A212722 a(n) = Sum_{k=0..n} n! * (1 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
%F A212722 Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F A212722 a(n,m) = Sum_{k=0..n} n! * m*(m + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
%F A212722 a(n) ~ n^(n-1) * (1+1/(2*c))^(n+1/2) / (2*sqrt(1+c) * exp(n) * c^n), where c = LambertW(1/sqrt(2)) = 0.450600515864833072257... . - _Vaclav Kotesovec_, Jul 15 2014
%e A212722 E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ...
%e A212722 such that, by definition:
%e A212722 log(A(x))/x = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6 + x^4*A(x)^8 + ...
%e A212722 Related expansions:
%e A212722 log(A(x)) = x/(1-x*A(x)^2) = x + 2*x^2/2! + 18*x^3/3! + 216*x^4/4! + 3640*x^5/5! + 78000*x^6/6! + 2032464*x^7/7! + 62400128*x^8/8! + ... + n*A366232(n-1)*x^n/n! + ...
%e A212722 A(x)^2 = 1 + 2*x + 8*x^2/2! + 68*x^3/3! + 880*x^4/4! + 15312*x^5/5! + 336064*x^6/6! +...
%e A212722 A(x)^4 = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3232*x^4/4! + 59104*x^5/5! + 1343296*x^6/6! +...
%t A212722 Table[Sum[n! * (1 + 2*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 15 2014 *)
%o A212722 (PARI) {a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+2*(n-k))^(k-1)*binomial(n-1, n-k)))}
%o A212722 (PARI) {a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^2+x*O(x^n)))); n!*polcoeff(A^m, n)}
%o A212722 for(n=0,21,print1(a(n),", "))
%Y A212722 Cf. A161630, A212917, A245265.
%Y A212722 Cf. A366232 (log).
%K A212722 nonn
%O A212722 0,3
%A A212722 _Paul D. Hanna_, May 25 2012