OFFSET
0,3
FORMULA
a(n) = Sum_{k=0..n} binomial(n - k, k)*(2*n)^k.
a(n) = A350470(n, n).
From Vaclav Kotesovec, Jan 08 2024: (Start)
a(n) = ((1 + sqrt(8*n+1))^(n+1) - (1 - sqrt(8*n+1))^(n+1)) / (sqrt(8*n+1) * 2^(n+1)).
a(n) ~ exp(sqrt(n/2)/2) * 2^(n/2 - 1) * n^(n/2) * (1 + 47/(96*sqrt(2*n))). (End)
MATHEMATICA
Table[Hypergeometric2F1[(1 - n)/2, -n/2, -n, -8 n ], {n, 0, 23}]
Table[FullSimplify[((1 + Sqrt[8*n + 1])^(n+1) - (1 - Sqrt[8*n + 1])^(n+1)) / (Sqrt[8*n + 1] * 2^(n+1))], {n, 0, 25}] (* Vaclav Kotesovec, Jan 08 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Mar 19 2022
STATUS
approved