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%I #24 Feb 15 2024 04:19:16
%S 1,2,8,38,200,1122,6576,39790,246672,1558658,10001592,64997814,
%T 426922392,2829624514,18901301984,127115260894,859978039840,
%U 5848754717314,39964745880552,274231943135686,1888891689752680,13055393137141282,90517646431869328
%N Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2) ).
%H P. Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(3*n-2*k+1,n-2*k).
%F a(n) = (1/(n+1)) *[x^n] ( 1/(1-x)^2 * (1+x^2) )^(n+1). - _Seiichi Manyama_, Feb 14 2024
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^2))/x)
%o (PARI) a(n, s=2, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);
%Y Cf. A218045, A369263.
%Y Cf. A052709, A109081.
%Y Cf. A370242.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jan 18 2024
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