A394186 - OEIS

%I #50 Jun 27 2026 16:28:42

%S 1,3,19,143,1151,9615,82223,714495,6281407,55711103,497534335,

%T 4467992831,40306810623,365002022143,3315975217919,30208418512895,

%U 275858397735935,2524389197707263,23143624647383039,212530950189522943,1954580341766602751,17999540908224323583

%N a(n) = Sum_{k=0..n} binomial(n+k,2*k) * (binomial(n+2*k,k) - binomial(n+2*k,k-1)).

%F a(n) = A071945(2*n,n).

%F a(n) = A243116(n) - A397383(n).

%F For m >= 0 and any constants r, s, define a_{m,r,s}(n) = Sum_{k=0..n} binomial(n+(m-1)*k,m*k) * (r*binomial(n+m*k,k) - s*binomial(n+m*k,k-1)). A_{m,r,s}(x) = Sum_{n>=0} a_{m,r,s}(n)*x^n = t*(1+t^m)*(r+(r+s)*t^(m-1)-s*t^m)/((1+t^(m-1))*(1+(m+1)*t^m-m*t^(m+1))), where t = t(x) satisfies t = 1 + x*t*(1+t^m).

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

%Y Cf. A071945, A243116, A346626, A397383.

%Y Cf. A116363, A395801, A396286.

%K nonn,changed

%O 0,2

%A _Seiichi Manyama_, Jun 22 2026