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Expansion of (1/x) * Series_Reversion( x * ((1-x)^2 + x^3) ).
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%I #10 Mar 23 2024 10:59:33

%S 1,2,7,29,131,623,3064,15423,78936,408958,2137993,11252163,59508232,

%T 315786764,1679410076,8941421014,47613443433,253359512287,

%U 1346009853489,7133000408765,37669665812955,198034693198875,1035095172883710,5371011415598595,27615259784888724

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

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-3*k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2+x^3))/x)

%o (PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(n+k, k)*binomial(3*n-k+1, n-3*k))/(n+1);

%Y Cf. A369214.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 23 2024