A394186 a(n) = Sum_{k=0..n} binomial(n+k,2*k) * (binomial(n+2*k,k) - binomial(n+2*k,k-1)).

1, 3, 19, 143, 1151, 9615, 82223, 714495, 6281407, 55711103, 497534335, 4467992831, 40306810623, 365002022143, 3315975217919, 30208418512895, 275858397735935, 2524389197707263, 23143624647383039, 212530950189522943, 1954580341766602751, 17999540908224323583

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

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

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

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).

Prog

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

Crossrefs

Cf. A071945, A243116, A346626, A397383.

Cf. A116363, A395801, A396286.

Sequence in context: A110525 A331716 A058859 * A291964 A333094 A095002

Adjacent sequences: A394183 A394184 A394185 * A394187 A394188 A394189

Keyword

nonn,changed

Author

Seiichi Manyama, Jun 22 2026

Status

approved