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%I #12 Nov 28 2017 00:16:03
%S 1,1,3,16,97,645,4539,33242,250715,1934131,15190377,121050779,
%T 976334857,7954909796,65378035310,541346845867,4511820592102,
%U 37819912868231,318639423484669,2696819879180630,22918021992024063,195480539889732302,1672951468057552136,14361120982757852887,123625070638172667688,1066939084577136857174
%N G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).
%H Paul D. Hanna, <a href="/A295810/b295810.txt">Table of n, a(n) for n = 0..520</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = 1 + Series_Reversion( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5) ).
%F (2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^4 + x^5).
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 97*x^4 + 645*x^5 + 4539*x^6 + 33242*x^7 + 250715*x^8 + 1934131*x^9 + 15190377*x^10 + 121050779*x^11 + 976334857*x^12 +...
%e such that A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).
%e RELATED SERIES.
%e A(x)^3 = 1 + 3*x + 12*x^2 + 67*x^3 + 423*x^4 + 2880*x^5 + 20607*x^6 + 152763*x^7 + 1162908*x^8 + 9037195*x^9 + 71398917*x^10 +...
%e A(x)^5 = 1 + 5*x + 25*x^2 + 150*x^3 + 990*x^4 + 6936*x^5 + 50640*x^6 + 381070*x^7 + 2934665*x^8 + 23016905*x^9 + 183216323*x^10 +...
%e A(x)^7 = 1 + 7*x + 42*x^2 + 273*x^3 + 1890*x^4 + 13671*x^5 + 102123*x^6 + 781838*x^7 + 6102530*x^8 + 48382012*x^9 + 388548244*x^10 +...
%e Series_Reversion(A(x) - 1) = x - 3*x^2 + 2*x^3 + 8*x^4 - 21*x^5 + 4*x^6 + 74*x^7 - 137*x^8 - 59*x^9 + 623*x^10 - 797*x^11 - 1083*x^12 + 4840*x^13 - 3793*x^14 - 12355*x^15 +...
%e which equals x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5).
%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^5 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A295808. A259757, A295537, A295538.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 27 2017
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