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%I #6 Feb 08 2021 16:08:56
%S 1,1,4,42,648,13620,362520,11696160,443748480,19362566160,
%T 955384416000,52602602199840,3197371241877120,212668103188981440,
%U 15364762939152506880,1198224396270059558400,100322035878465399398400,8975632678570151712518400,854594260998625751469465600
%N E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*A(x)^(2*n)*exp(x*A(x)^n)/n! and Q(x) = Sum_{n>=0} x^n*A(x)^n*exp(x*A(x)^(n+1))/n!.
%F E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where
%F P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^(2*n) * exp(x*A(x)^n) / n!,
%F Q(x) = Sum_{n>=0} x^n * A(x)^n * exp(x*A(x)^(n+1)) / n!.
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 42*x^3/3! + 648*x^4/4! + 13620*x^5/5! + 362520*x^6/6! + 11696160*x^7/7! + 443748480*x^8/8! + 19362566160*x^9/9! + ...
%e such that A(x) = P(x)/Q(x) where
%e P(x) = exp(x) + 2*x*A(x)^2*exp(x*A(x)) + 3*x^2*A(x)^4*exp(x*A(x)^2)/2! + 4*x^3*A(x)^6*exp(x*A(x)^3)/3! + 5*x^4*A(x)^8*exp(x*A(x)^4)/4! + ...
%e Q(x) = exp(x*A(x)) + x*A(x)*exp(x*A(x)^2) + x^2*A(x)^2*exp(x*A(x)^3)/2! + x^3*A(x)^3*exp(x*A(x)^4)/3! + x^4*A(x)^4*exp(x*A(x)^5)/4! + ...
%e explicitly,
%e P(x) = 1 + 3*x + 16*x^2/2! + 152*x^3/3! + 2256*x^4/4! + 46172*x^5/5! + 1207456*x^6/6! + 38466192*x^7/7! + 1445453344*x^8/8! + 62597927312*x^9/9! + ...
%e Q(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 832*x^4/4! + 16072*x^5/5! + 405304*x^6/6! + 12590216*x^7/7! + 464416544*x^8/8! + 19828679264*x^9/9! + ...
%o (PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1);
%o for(i=0, n,
%o P = sum(m=0, n, (m+1)*x^m*A^(2*m)/m! * exp(x*A^m + x*O(x^n)) );
%o Q = sum(m=0, n, x^m*A^m/m! * exp(x*A^(m+1) + x*O(x^n)) );
%o A = P/Q); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A340942, A340938.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 08 2021
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