A357502 a(n) = ((1 + sqrt(n))^n - (1 - sqrt(n))^n)/(2*sqrt(n)).

1, 2, 6, 20, 80, 342, 1624, 8136, 43776, 246410, 1463264, 9033180, 58200064, 387905182, 2679200640, 19068105488, 139929124864, 1054773505170, 8167509816832, 64795371984420, 526534098026496, 4374163243287398, 37135913476691968, 321727849480560600

Offset: 1

Author

Alexander R. Povolotsky, Oct 01 2022

Keywords

• nonn
• easy

Crossrefs

Main diagonal of A099173.

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):
seq(a(n), n=1..26);  # Alois P. Heinz, Oct 01 2022

Mathematica

Simplify[Table[((1 + Sqrt[n])^n - (1 - Sqrt[n])^n)/(2*Sqrt[n]),{n,26}]] (* Stefano Spezia, Oct 03 2022 *)

PARI

a(n) = polcoeff(lift( Mod('x,'x^2-2*'x+1-n)^n ),1); \\ Kevin Ryde, Oct 01 2022

Python

from sympy import simplify, sqrt
def A357502(n): return simplify(((1+sqrt(n))**n-(1-sqrt(n))**n)/sqrt(n))>>1 # Chai Wah Wu, Oct 14 2022

Formula

From Alois P. Heinz, Oct 01 2022: (Start)

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

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

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