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A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ). 4

%I

%S 1,1,3,25,313,5341,115651,3036517,93767185,3330162073,133737097411,

%T 5992748728561,296433923379529,16044427276953973,943207466055927619,

%U 59848531677741706621,4076826825898115406241,296742863575079244130225

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

%H G. C. Greubel, <a href="/A212722/b212722.txt">Table of n, a(n) for n = 0..358</a>

%H Vaclav Kotesovec, <a href="http://oeis.org/A245265/a245265.pdf">Asymptotic of sequences A161630, A212722, A212917 and A245265</a>

%F a(n) = Sum_{k=0..n} n! * (1 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

%F a(n,m) = Sum_{k=0..n} n! * m*(m + 2*(n-k))^(k-1)/k! * C(n-1,n-k).

%F 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 E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! +...

%e such that, by definition:

%e 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 Related expansions:

%e 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! +...

%e 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 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 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 (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 (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 for(n=0,21,print1(a(n),", "))

%Y Cf. A161630, A212917, A245265.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 25 2012

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Last modified May 9 02:50 EDT 2021. Contains 343685 sequences. (Running on oeis4.)