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%I #11 Apr 07 2018 22:40:32
%S 1,1,6,60,796,12873,243648,5274630,128693820,3501032280,105278025690,
%T 3473075317080,124851287129604,4860913404292030,203839755469977840,
%U 9161170785398640570,439317870410796482460,22390147212903891054540,1208526686930226056003640,68866979112831866042953128,4131396903316522288744806384,260262349821990852529147365849
%N G.f. A(x) satisfies: A(x) = 1 + x * (x*A(x)^4)' / (x*A(x))'.
%C Compare to: C(x) = 1 + x * (x*C(x)^2)' / (x*C(x))' holds when C(x) = 1 + x*C(x)^2, which is a g.f. of the Catalan numbers (A000108).
%C Compare to: G(x) = 1 + x * (x*G(x)^3)' / (x*G(x))' holds when G(x) = 1/(1 - x*G(x)/(1 - 2*x*G(x)/(1 - 3*x*G(x)/(1 - ...)))), a continued fraction, which is the g.f. of A301363.
%F G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).
%e G.f.: A(x) = 1 + x + 6*x^2 + 60*x^3 + 796*x^4 + 12873*x^5 + 243648*x^6 + 5274630*x^7 + 128693820*x^8 + 3501032280*x^9 + 105278025690*x^10 + ...
%e such that A(x) = 1 + x * (x*A(x)^4)' / (x*A(x))'.
%o (PARI) /* Differential Equation */
%o {a(n) = my(A=1); for(i=0,n, A = 1 + x*(x*A^4)'/(x*A +x^2*O(x^n))'); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n),", "))
%Y Cf. A301363, A302101.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 07 2018
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