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%I #5 Feb 09 2021 02:44:53
%S 1,1,2,12,96,1120,16260,290640,6108480,148353408,4081855680,
%T 125613124560,4274457264000,159409774592640,6465790781049600,
%U 283412493387223200,13350812617606464000,672683432660494295040,36100038651180773068800
%N E.g.f. A(x) satisfies: Sum_{n>=0} x^n * exp(x*A(x)^n) / n! = exp(x*A(x) + x/A(x)).
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 1120*x^5/5! + 16260*x^6/6! + 290640*x^7/7! + 6108480*x^8/8! + 148353408*x^9/9! + ...
%e where
%e exp(x*A(x) + x/A(x)) = exp(x) + x*exp(x*A(x)) + x^2*exp(x*A(x)^2)/2! + x^3*exp(x*A(x)^3)/3! + x^4*exp(x*A(x)^4)/4! + x^5*exp(x*A(x)^5)/5! +...
%e explicitly,
%e exp(x*A(x) + x/A(x)) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 88*x^4/4! + 872*x^5/5! + 11464*x^6/6! + 189968*x^7/7! + 3774208*x^8/8! + 87674336*x^9/9! + ...
%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 97*x^4/4! + 981*x^5/5! + 13291*x^6/6! + 222013*x^7/7! + 4458273*x^8/8! + 104169817*x^9/9! + ...
%o (PARI) {a(n) = my(A=1+x +x^3*O(x^n), H=A);
%o for(k=1, n, A = A*exp(-x*A)*exp(-x/A) * sum(m=0, n+3, x^m/m! * exp(x*A^m +x^3*O(x^n)) );
%o A = truncate( H + polcoeff(A, k+2)*x^k ) +x^3*O(x^n); H=A); n!*polcoeff(W=A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A340941, A340939.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 08 2021
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