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

%I #20 Apr 14 2024 08:50:06

%S 1,7,67,741,8909,113107,1492103,20251945,280978681,3967031839,

%T 56811348235,823250855181,12049087175493,177857857845675,

%U 2644773866954255,39581787842355409,595745692419162737,9011736489133233463,136932249972928786387

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

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

%Y Essentially the same as A243675.

%Y Cf. A034015.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 21 2024