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A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).

%I A369226 #33 Feb 15 2024 04:38:51
%S A369226 1,1,4,13,53,220,968,4373,20271,95705,458904,2228220,10934524,
%T A369226 54143848,270189008,1357428997,6860264323,34853234867,177900211204,
%U A369226 911867479717,4691701977973,24222505191984,125448280976224,651555603531308,3392951906596708,17711433386188300
%N A369226 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ).
%H A369226 Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H A369226 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A369226 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+2,k) * binomial(2*n-2*k,n-2*k).
%F A369226 a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^2 )^(n+1). - _Seiichi Manyama_, Feb 14 2024
%o A369226 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+x^2)^2)/x)
%o A369226 (PARI) a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
%Y A369226 Cf. A052709, A369262.
%Y A369226 Cf. A218045, A370243.
%K A369226 nonn
%O A369226 0,3
%A A369226 _Seiichi Manyama_, Jan 18 2024