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Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x^2)^2) ).
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%I #9 Jan 23 2024 11:14:33

%S 1,1,3,9,30,107,396,1513,5915,23554,95202,389555,1610588,6717816,

%T 28234064,119452553,508330809,2174393331,9343913933,40319400738,

%U 174630125428,758916134002,3308320668768,14462616815619,63388694309005,278492994845776,1226241871745376

%N Expansion of (1/x) * Series_Reversion( x / ((1+x) * (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(n+1,n-2*k).

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

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

%Y Cf. A036765, A369441.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 23 2024