A371391 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+2*x)^2 ).
1, 5, 34, 269, 2326, 21314, 203428, 2000957, 20142862, 206524790, 2149261852, 22644243218, 241061343004, 2589022298084, 28019201644744, 305254481274269, 3345077342003134, 36846738570089774, 407754101877613804, 4531049315843043974, 50538820796852529364
Offset: 0
Keywords
Programs
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Mathematica
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 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)/(1+2*x)^2)/x) -
PARI
a(n) = sum(k=0, n, 2^k*binomial(2*(n+1), k)*binomial(2*n-k, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(2*(n+1),k) * binomial(2*n-k,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+2*x)^2 / (1-x) )^(n+1). - Seiichi Manyama, Jul 31 2025
a(n) ~ 2^(2*n-2) * 3^(n+2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2025