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Expansion of (1/x) * Series_Reversion( x * (1-x+x^3)^2 ).
4

%I #14 Jan 13 2024 10:45:49

%S 1,2,7,28,121,546,2531,11934,56867,272580,1309505,6285630,30057195,

%T 142754008,671062828,3108766166,14108600499,62170980416,262108536781,

%U 1027886900446,3509371721163,8204350476210,-12172347463045,-361684831407060,-3497893818262311

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

%H Seiichi Manyama, <a href="/A368970/b368970.txt">Table of n, a(n) for n = 0..1000</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/3)} (-1)^k * binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-3*k).

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

%o (PARI) a(n, s=3, t=2, u=0) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

%Y Cf. A063033, A368962.

%Y Cf. A368974, A368976.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 10 2024