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

%I #10 Mar 03 2024 09:52:48

%S 1,1,2,5,13,35,96,264,719,1914,4888,11583,23540,29907,-57120,-695742,

%T -3938275,-18266657,-76983710,-306116660,-1168987469,-4327878214,

%U -15617536020,-55097816085,-190320077663,-643865817117,-2131713980560,-6893257768141

%N Expansion of (1/x) * Series_Reversion( x * (1/(1-x^4) - x) ).

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

%F a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(2*n-4*k+1,k) * binomial(2*n-4*k,n-4*k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1/(1-x^4)-x))/x)

%o (PARI) a(n) = sum(k=0, n\4, (-1)^k*binomial(2*n-4*k+1, k)*binomial(2*n-4*k, n-4*k))/(n+1);

%Y Cf. A370839, A370840.

%Y Cf. A369631.

%K sign

%O 0,3

%A _Seiichi Manyama_, Mar 03 2024