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A274379 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).

%I A274379 #20 Jan 28 2024 07:26:28
%S A274379 1,3,13,70,429,2842,19794,142758,1056655,7980280,61251261,476387379,
%T A274379 3746317414,29738316330,237968639936,1917578268288,15546796822656,
%U A274379 126728260011920,1037987924978125,8538459191677170,70509828893263474,584310452973463242,4857624566855734836,40501472981905806550,338594135314564168494,2837641019938074131463,23835438376045780734390,200633658871150345742269,1692132786239339256115050,14297391426538004065333910,121009206594941545408186768
%N A274379 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).
%H A274379 Seiichi Manyama, <a href="/A274379/b274379.txt">Table of n, a(n) for n = 0..1000</a>
%F A274379 G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^3) / (1+x)^3 ).
%F A274379 G.f. satisfies: A( x*(1 - x^2*(1+x)^3)/(1+x)^3 ) = (1+x)^3/(1 - x^2*(1+x)^3).
%F A274379 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n+3*k+3,n-2*k). - _Seiichi Manyama_, Jan 27 2024
%e A274379  G.f.: A(x) = 1 + 3*x + 13*x^2 + 70*x^3 + 429*x^4 + 2842*x^5 + 19794*x^6 + 142758*x^7 + 1056655*x^8 + 7980280*x^9 + ...
%e A274379 such that A(x) = 1 + 3*x*A(x) + x^2*(3*A(x)^2 + A(x)^3) + x^3*(A(x)^3 + 3*A(x)^4) + 3*x^4*A(x)^5 + x^5*A(x)^6.
%o A274379 (PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^3 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}
%o A274379 for(n=0, 30, print1(a(n), ", "))
%o A274379 (PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A, n)}
%o A274379 for(n=0, 30, print1(a(n), ", "))
%Y A274379 Cf. A181734, A274378.
%Y A274379 Cf. A274735, A369600.
%K A274379 nonn
%O A274379 0,2
%A A274379 _Paul D. Hanna_, Aug 04 2016