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A349011 G.f. A(x) satisfies: A(x) = (1 - x * A(-x)) / (1 - 2 * x * A(x)).

%I A349011 #7 Aug 19 2022 05:06:38
%S A349011 1,1,5,17,105,433,2925,13185,93425,443009,3233205,15840209,117950745,
%T A349011 591187953,4466545245,22766535297,173906505825,897941153665,
%U A349011 6918379345125,36089242700049,279988660639305,1472715584804529,11490841104036045,60857608450349313,477104721264920145
%N A349011 G.f. A(x) satisfies: A(x) = (1 - x * A(-x)) / (1 - 2 * x * A(x)).
%F A349011 a(0) = 1; a(n) = (-1)^n * a(n-1) + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).
%p A349011 A349011 := proc(n)
%p A349011     option remember ;
%p A349011     if n = 0 then
%p A349011         1;
%p A349011     else
%p A349011         (-1)^n*procname(n-1)+2*add(procname(k)*procname(n-k-1),k=0..n-1) ;
%p A349011     end if;
%p A349011 end proc:
%p A349011 seq(A349011(n),n=0..40) ; # _R. J. Mathar_, Aug 19 2022
%t A349011 nmax = 24; A[_] = 0; Do[A[x_] = (1 - x A[-x])/(1 - 2 x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349011 a[0] = 1; a[n_] := a[n] = (-1)^n a[n - 1] + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
%Y A349011 Cf. A001003, A103210, A348957.
%K A349011 nonn
%O A349011 0,3
%A A349011 _Ilya Gutkovskiy_, Nov 05 2021