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A364374 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^2).

%I A364374 #26 Sep 08 2024 08:25:09
%S A364374 1,0,-1,1,2,-6,-1,28,-31,-98,288,131,-1730,1638,7431,-19583,-15502,
%T A364374 135642,-99523,-664050,1535896,1816196,-11902728,5944326,64487669,
%U A364374 -129346490,-213116764,1112382523,-277762230,-6572175490,11287106695,25078981772,-107983368519,-1826241850
%N A364374 G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^2).
%H A364374 Seiichi Manyama, <a href="/A364374/b364374.txt">Table of n, a(n) for n = 0..1000</a>
%F A364374 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1).
%F A364374 D-finite with recurrence 15*n*(n+1)*a(n) +2*n*(13*n-11)*a(n-1) +12*(9*n^2-19*n+9)*a(n-2) +2*(10*n^2-65*n+99)*a(n-3) -4*(n-3)*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jul 25 2023
%F A364374 A(x) = (1/x) * series_reversion(x*(1 + x + x^2)/(1 + x)). - _Peter Bala_, Sep 08 2024
%p A364374 A364374 := proc(n)
%p A364374     add( (-1)^k*binomial(n+k+1,k) * binomial(n+k+1,n-k)/(n+k+1),k=0..n) ;
%p A364374 end proc:
%p A364374 seq(A364374(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%t A364374 nmax = 33;
%t A364374 A[_] = 1;
%t A364374 Do[A[x_] = (1+x*A[x])*(1-x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
%t A364374 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 21 2023 *)
%o A364374 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k+1, k)*binomial(n+k+1, n-k)/(n+k+1));
%Y A364374 Cf. A364375, A364376.
%Y A364374 Cf. A057078, A007863, A364371.
%K A364374 sign,easy
%O A364374 0,5
%A A364374 _Seiichi Manyama_, Jul 21 2023