A366025 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x^5) ).
1, 1, 2, 5, 14, 43, 139, 465, 1595, 5577, 19804, 71228, 258946, 950030, 3513050, 13079920, 48993149, 184490361, 698020080, 2652192675, 10115878915, 38717526745, 148655862210, 572412768275, 2209969761924, 8553073927858, 33176952295730, 128960722306128
Offset: 0
Keywords
Programs
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Mathematica
CoefficientList[InverseSeries[Series[x(1-x)/(1+x^5),{x,0,28}],x]/x,x] (* Stefano Spezia, Sep 26 2023 *) -
PARI
a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(2*n-5*k+1, n-4*k)/(2*n-5*k+1));
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PARI
Vec(serreverse(x*(1-x)/(1+x^5)+O(x^30))/x) \\ Michel Marcus, Sep 26 2023
Formula
G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^4*A(x)^3).
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k,k) * binomial(2*n-5*k+1,n-4*k)/(2*n-5*k+1) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n+1,k) * binomial(2*n-5*k,n-5*k).