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Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2)^2) ).
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%I #10 Jan 23 2024 11:14:29

%S 1,2,7,30,141,704,3666,19686,108222,606062,3445308,19829680,115323955,

%T 676659960,4000719012,23811922678,142557391306,857894530348,

%U 5186614665121,31487226410770,191871141682557,1173163962971056,7195329233469552,44255915928488880

%N Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2)^2) ).

%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(2*n+2,k) * binomial(2*n+2,n-2*k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x^2)^2))/x)

%o (PARI) a(n, s=2, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);

%Y Cf. A369440.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 23 2024