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A274735 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x*A(x)^2).

%I A274735 #21 Jan 28 2024 09:19:22
%S A274735 1,4,26,210,1901,18445,187524,1971672,21263360,233907762,2614446624,
%T A274735 29607343948,338977591904,3917185497535,45629006313280,
%U A274735 535199773167207,6315789123860388,74932400322972992,893276792585933870,10694510040508714014,128531711285410216883,1550159476645634696615,18755239991772817629972,227577929298568261967650,2768820313297861609739979
%N A274735 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x*A(x)^2).
%C A274735 More generally, if G(x) satisfies
%C A274735 G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
%C A274735 G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).
%H A274735 Seiichi Manyama, <a href="/A274735/b274735.txt">Table of n, a(n) for n = 0..903</a>
%F A274735 G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^3) / (1+x)^3 ).
%F A274735 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(3*n+3*k+3,n-k). - _Seiichi Manyama_, Jan 27 2024
%e A274735 G.f.: A(x) = 1 + 4*x + 26*x^2 + 210*x^3 + 1901*x^4 + 18445*x^5 + 187524*x^6 + 1971672*x^7 + 21263360*x^8 +...
%o A274735 (PARI) {a(n) = my(A=1); for(i=1,n, A = (1 + x*A)^3 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A,n)}
%o A274735 for(n=0,30,print1(a(n),", "))
%o A274735 (PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A,n)}
%o A274735 for(n=0,30,print1(a(n),", "))
%Y A274735 Cf. A007863, A274734.
%Y A274735 Cf. A274379, A369600.
%K A274735 nonn
%O A274735 0,2
%A A274735 _Paul D. Hanna_, Aug 02 2016