|
%I #5 Feb 25 2021 02:29:12
%S 1,1,1,7,37,501,5491,100423,1750729,40959145,983720071,28320815931,
%T 862784739037,29728071447517,1093233588351451,44072354567202991,
%U 1894875749109296401,87722973117476411985,4313507974583583130255,225835795120784011395427
%N E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (2^n*x^n/n!) * exp(x*A(x)^n) and Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n).
%F E.g.f. A(x) satisfies:
%F (1) A(x) = P(x)/Q(x) where
%F P(x) = Sum_{n>=0} (2^n*x^n/n!) * exp(x*A(x)^n) and
%F Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n).
%F (2) A(x) = P(x)/Q(x) where
%F P(x) = Sum_{n>=0} (x^n/n!) * exp(2*x*A(x)^n) and
%F Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n).
%e E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 37*x^4/4! + 501*x^5/5! + 5491*x^6/6! + 100423*x^7/7! + 1750729*x^8/8! + 40959145*x^9/9! + 983720071*x^10/10! + ...
%e such that A(x) = P(x)/Q(x) where
%e P(x) = exp(x) + 2*x*exp(x*A(x)) + 2^2*x^2*exp(x*A(x)^2)/2! + 2^3*x^3*exp(x*A(x)^3)/3! + 2^4*x^4*exp(x*A(x)^4)/4! + 2^5*x^5*exp(x*A(x)^5)/5! + ...
%e Q(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 also
%e P(x) = exp(x) + x*exp(2*x*A(x)) + x^2*exp(2*x*A(x)^2)/2! + x^3*exp(2*x*A(x)^3)/3! + x^4*exp(2*x*A(x)^4)/4! + x^5*exp(2*x*A(x)^5)/5! + ...
%e explicitly,
%e P(x) = 1 + 3*x + 9*x^2/2! + 39*x^3/3! + 249*x^4/4! + 2323*x^5/5! + 27789*x^6/6! + 421851*x^7/7! + 7577425*x^8/8! + 160651683*x^9/9! + 3868078869*x^10/10! + ...
%e Q(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 76*x^4/4! + 652*x^5/5! + 7054*x^6/6! + 102650*x^7/7! + 1741016*x^8/8! + 35911592*x^9/9! + 832884154*x^10/10! + ...
%o (PARI) {a(n) = my(A=1 +x*O(x^n),P=1,Q=1); for(i=1, n,
%o P = sum(m=0, n, 2^m*x^m/m! * exp(x*A^m +x*O(x^n)) );
%o Q = sum(m=0, n, x^m/m! * exp(x*A^m +x*O(x^n)) );
%o A = P/Q); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1 +x*O(x^n),P=1,Q=1); for(i=1, n,
%o P = sum(m=0, n, x^m/m! * exp(2*x*A^m +x*O(x^n)) );
%o Q = sum(m=0, n, x^m/m! * exp(x*A^m +x*O(x^n)) );
%o A = P/Q); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%K nonn
%O 0,4
%A _Paul D. Hanna_, Feb 22 2021
| 