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Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).
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%I #18 Feb 15 2024 04:16:15

%S 1,2,10,54,329,2126,14356,100030,713956,5193064,38354066,286860714,

%T 2168308302,16537766036,127114940840,983657456878,7657060437148,

%U 59917814944376,471062428422152,3718952705982232,29471640802526185,234356062245289566,1869405604537134116

%N Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

%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(3*n+3,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)^3 )^(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)^3)/x)

%o (PARI) a(n, s=2, t=3, 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, A369208.

%Y Cf. A369262, A369264.

%Y Cf. A370245.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 18 2024