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Expansion of (1/x) * Series_Reversion( x / ( (1+x) * (1+3*x)^3 ) ).
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%I #9 Mar 22 2024 09:00:15

%S 1,10,136,2134,36379,654670,12239560,235407070,4627854244,92576970280,

%T 1878395043232,38564373070090,799651963174978,16722655896174004,

%U 352289843771100400,7469327989417602862,159263992188702829900,3412969567344634872952

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

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

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+3*x)^3))/x)

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

%Y Cf. A364923.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 21 2024