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A291817 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x + x*A(x).

%I A291817 #3 Sep 02 2017 03:53:18
%S A291817 1,4,40,580,10312,209752,4707952,114128308,2946787192,80268150808,
%T A291817 2290811949904,68149759850680,2105074730357968,67308231895605520,
%U A291817 2222306773263886624,75615701295449074084,2647156154616207962920,95215874465318554556776,3514739264129342710455184,133012394550946993742673304,5156112210399927390763631056,204570581814658024128260509360
%N A291817 G.f. A(x) satisfies: A(x - 3*x*A(x)) = x + x*A(x).
%F A291817 G.f. A(x) also satisfies:
%F A291817 (1) A(x) = (4/3)*Series_Reversion( x - 3*x*A(x) ) - x/3.
%F A291817 (2) A( (3*A(x) + x)/4 ) = (A(x) - x) / (3*A(x) + x).
%F A291817 a(n) = Sum_{k=0..n-1} A291820(n, k) * 3^k * 4^(n-k-1).
%e A291817 G.f.: A(x) = x + 4*x^2 + 40*x^3 + 580*x^4 + 10312*x^5 + 209752*x^6 + 4707952*x^7 + 114128308*x^8 + 2946787192*x^9 + 80268150808*x^10 +...
%e A291817 such that  A(x - 3*x*A(x)) = x + x*A(x).
%e A291817 RELATED SERIES.
%e A291817 A(x - 3*x*A(x)) = x + x^2 + 4*x^3 + 40*x^4 + 580*x^5 + 10312*x^6 + 209752*x^7 + 4707952*x^8 +...
%e A291817 which equals x + x*A(x).
%e A291817 Series_Reversion( x - 3*x*A(x) ) = x + 3*x^2 + 30*x^3 + 435*x^4 + 7734*x^5 + 157314*x^6 + 3530964*x^7 + 85596231*x^8 +...
%e A291817 which equals (3/4)*A(x) + x/4.
%e A291817 A( (3*A(x) + x)/4 )  = x + 7*x^2 + 94*x^3 + 1651*x^4 + 33886*x^5 + 773458*x^6 + 19117780*x^7 + 503529979*x^8 + 13983485770*x^9 + 406470316978*x^10 +...
%e A291817 which equals (A(x) - x) / (3*A(x) + x).
%o A291817 (PARI) {a(n) = my(A=x); for(i=1, n, A = (4/3)*serreverse( x - 3*x*A +x*O(x^n) ) - x/3 ); polcoeff(A, n)}
%o A291817 for(n=1, 30, print1(a(n), ", "))
%Y A291817 Cf. A291820, A291813, A291814, A291815, A291816, A291818, A291819.
%K A291817 nonn
%O A291817 1,2
%A A291817 _Paul D. Hanna_, Sep 02 2017