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A274378 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x^2*A(x)^3).

%I A274378 #17 Jan 27 2024 10:31:53
%S A274378 1,2,6,24,111,552,2873,15458,85312,480314,2747845,15928080,93347153,
%T A274378 552181372,3292571913,19769887128,119430685503,725375643416,
%U A274378 4426786390959,27131644746326,166932630227613,1030684209393288,6383992918008611,39657230694169284,247008096338698523,1542292860296588558,9651791500807437834,60528789932966226468,380333245334293851637,2394179659042901060436,15096873553004201457425
%N A274378 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x^2*A(x)^3).
%H A274378 Seiichi Manyama, <a href="/A274378/b274378.txt">Table of n, a(n) for n = 0..1000</a>
%F A274378 G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^2) / (1+x)^2 ).
%F A274378 G.f. satisfies: A( x*(1 - x^2*(1+x)^2)/(1+x)^2 ) = (1+x)^2/(1 - x^2*(1+x)^2).
%F A274378 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(2*n+2*k+2,n-2*k). - _Seiichi Manyama_, Jan 27 2024
%e A274378  G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 111*x^4 + 552*x^5 + 2873*x^6 + 15458*x^7 + 85312*x^8 +...
%e A274378 such that A(x) = 1 + 2*x*A(x) + x^2*(A(x)^2 + A(x)^3) + 2*x^3*A(x)^4 + x^4*A(x)^5.
%o A274378 (PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^2 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}
%o A274378 for(n=0, 30, print1(a(n), ", "))
%o A274378 (PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A, n)}
%o A274378 for(n=0, 30, print1(a(n), ", "))
%Y A274378 Cf. A181734, A274379.
%K A274378 nonn
%O A274378 0,2
%A A274378 _Paul D. Hanna_, Aug 04 2016