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Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^2 ) ).
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%I #16 Dec 25 2024 09:17:04

%S 1,6,49,462,4734,51216,575705,6657846,78703438,946740132,11551512042,

%T 142616584380,1778372098000,22365031140900,283341912929865,

%U 3612782260978470,46326552943960278,597034029166804068,7728885814331709374,100458438481544424996

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

%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(2*(n+1),k) * binomial(2*(n+1),n-k).

%F a(n) = A219538(n+1)/2. - _Seiichi Manyama_, Dec 24 2024

%F a(n) = (1/(n+1)) * [x^n] ( (1+x) * (1+2*x) )^(2*(n+1)). - _Seiichi Manyama_, Dec 25 2024

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

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

%Y Cf. A219538, A371398.

%Y Cf. A379546, A379547.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 21 2024