A357580 a(n) = ((1 + sqrt(n))^n - (1 - sqrt(n))^n)/(2*n*sqrt(n)).

1, 1, 2, 5, 16, 57, 232, 1017, 4864, 24641, 133024, 752765, 4476928, 27707513, 178613376, 1191756593, 8231124992, 58598528065, 429868937728, 3239768599221, 25073052286976, 198825601967609, 1614604933769216, 13405327061690025, 113725655719346176

Offset: 1

Author

Alexander R. Povolotsky, Oct 04 2022

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..698

Crossrefs

Cf. A099173, A357502.

Programs

Maple

b:= proc(n, k) option remember;
     `if`(n<2, n, 2*b(n-1, k)+(k-1)*b(n-2, k))
    end:
a:= n-> b(n$2)/n:
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 04 2022

Mathematica

Expand[Table[((1 + Sqrt[n])^n - (1 - Sqrt[n])^n)/(2*n*Sqrt[n]), {n, 1, 27}]]

Python

from sympy import simplify, sqrt
def A357580(n): return simplify(((1+sqrt(n))**n-(1-sqrt(n))**n)/(n*sqrt(n)))>>1 # Chai Wah Wu, Oct 14 2022

Formula

a(n) = A357502(n)/n.

From Alois P. Heinz, Oct 04 2022: (Start)

a(n) = [x^n] x/(n*(1-2*x-(n-1)*x^2)).

a(n) = Sum_{j=0..floor(n/2)} n^(j-1) * binomial(n,2*j+1).

a(n) = A099173(n,n)/n. (End)