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A295809 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).

%I A295809 #8 Nov 28 2017 00:44:39
%S A295809 1,1,2,9,44,238,1363,8129,49947,313982,2009804,13054923,85835763,
%T A295809 570162938,3820449453,25792692527,175277931388,1198017908942,
%U A295809 8230391249548,56801549666858,393622890736512,2737841490099777,19107124975145342,133755761621788177,938960164858527807,6608463199584560132,46621379983243723382,329627663780846842009
%N A295809 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).
%C A295809 Compare to: G(x)^3 = 1+x + x*(G(x)^2 + G(x)^4) holds when G(x) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).
%H A295809 Paul D. Hanna, <a href="/A295809/b295809.txt">Table of n, a(n) for n = 0..520</a>
%F A295809 G.f. A(x) satisfies:
%F A295809 (1) A(x) = 1 + Series_Reversion( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5) ).
%F A295809 (2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^2 - x^4 + x^5).
%e A295809 G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 44*x^4 + 238*x^5 + 1363*x^6 + 8129*x^7 + 49947*x^8 + 313982*x^9 + 2009804*x^10 + 13054923*x^11 + 85835763*x^12 +...
%e A295809 such that A(x)^3 = 1+x + x*(A(x)^2 + A(x)^7).
%e A295809 RELATED SERIES.
%e A295809 A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 204*x^4 + 1125*x^5 + 6536*x^6 + 39390*x^7 + 243966*x^8 + 1543350*x^9 + 9929589*x^10 +...
%e A295809 A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 110*x^4 + 600*x^5 + 3459*x^6 + 20728*x^7 + 127824*x^8 + 805852*x^9 + 5170270*x^10 +...
%e A295809 A(x)^7 = 1 + 7*x + 35*x^2 + 182*x^3 + 1015*x^4 + 5936*x^5 + 35931*x^6 + 223238*x^7 + 1415526*x^8 + 9123737*x^9 + 59601227*x^10 +...
%o A295809 (PARI)  {a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^2 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
%o A295809 for(n=0, 30, print1(a(n), ", "))
%o A295809 (PARI) {a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 2*x + 5*x^2 + 6*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
%o A295809 for(n=0, 30, print1(a(n), ", "))
%Y A295809 Cf. A295808, A295810. A259757, A295537, A295538.
%K A295809 nonn
%O A295809 0,3
%A A295809 _Paul D. Hanna_, Nov 27 2017