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%I #22 Oct 08 2023 06:53:46
%S 1,1,3,31,469,9681,254701,8131999,305626329,13218345793,646712664121,
%T 35315446759671,2129341219106773,140506900034640049,
%U 10071589943109973461,779311468200041101711,64742128053980794659121,5747587082198264156035329,543023929087191507383612785
%N E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).
%C From _Vaclav Kotesovec_, Jul 15 2014: (Start)
%C Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
%C r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
%C A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
%C a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
%C (End)
%H G. C. Greubel, <a href="/A212917/b212917.txt">Table of n, a(n) for n = 0..349</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 + 3*(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 + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
%F a(n) ~ 3^(n-2/3) * n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2/3) * exp(n) * c^(2*n+1/6)), where c = LambertW(sqrt(3)/2) = 0.5166154518588324282494... . - _Vaclav Kotesovec_, Jul 15 2014
%e E.g.f: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + ...
%e such that, by definition:
%e log(A(x))/x = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9 + x^4*A(x)^12 + ...
%e Related expansions:
%e log(A(x)) = x/(1-x*A(x)^3) = x + 2*x^2/2! + 24*x^3/3! + 348*x^4/4! + 7140*x^5/5! + 186750*x^6/6! + ... + n*A366233(n-1)*x^n/n! + ...
%e A(x)^3 = 1 + 3*x + 15*x^2/2! + 153*x^3/3! + 2421*x^4/4! + 51363*x^5/5! + 1375029*x^6/6! + ...
%e A(x)^6 = 1 + 6*x + 48*x^2/2! + 576*x^3/3! + 9864*x^4/4! + 221256*x^5/5! + 6156756*x^6/6! + ...
%t Table[Sum[n! * (1 + 3*(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+3*(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^3+x*O(x^n)))); n!*polcoeff(A^m, n)}
%o for(n=0, 21, print1(a(n), ", "))
%Y Cf. A161630, A212722, A245265.
%Y Cf. A366233 (log).
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 30 2012
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