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A371391 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^2 ).

%I A371391 #18 Jul 31 2025 09:42:57
%S A371391 1,5,34,269,2326,21314,203428,2000957,20142862,206524790,2149261852,
%T A371391 22644243218,241061343004,2589022298084,28019201644744,
%U A371391 305254481274269,3345077342003134,36846738570089774,407754101877613804,4531049315843043974,50538820796852529364
%N A371391 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^2 ).
%H A371391 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A371391 a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(2*(n+1),k) * binomial(2*n-k,n-k).
%F A371391 a(n) = (1/(n+1)) * [x^n] ( (1+2*x)^2 / (1-x) )^(n+1). - _Seiichi Manyama_, Jul 31 2025
%F A371391 a(n) ~ 2^(2*n-2) * 3^(n+2) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 31 2025
%t A371391 Table[Sum[2^k*Binomial[2*(n+1), k]*Binomial[2*n-k, n-k]/(n+1), {k,0,n}], {n,0,30}] (* _Vaclav Kotesovec_, Jul 31 2025 *)
%o A371391 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+2*x)^2)/x)
%o A371391 (PARI) a(n) = sum(k=0, n, 2^k*binomial(2*(n+1), k)*binomial(2*n-k, n-k))/(n+1);
%Y A371391 Cf. A047891, A371392.
%K A371391 nonn
%O A371391 0,2
%A A371391 _Seiichi Manyama_, Mar 21 2024