A387459 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k), where i is the imaginary unit.

1, 2, 11, 96, 1121, 16280, 281987, 5666304, 129488641, 3315041568, 93958705499, 2920298135040, 98749216968481, 3608920706225536, 141743544911838547, 5953777300691189760, 266315973364196014081, 12638365012375994704384, 634207216217264733599531, 33552879853099295377612800

Offset: 0

Author

Vaclav Kotesovec, Aug 29 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A387430.

Programs

Magma

C := ComplexField(); [Floor(Abs( ((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2)): n in [0..30]]; // Vincenzo Librandi, Aug 30 2025

Mathematica

Table[Sum[(n-I)^k*(n+I)^(n-k), {k, 0, n}], {n, 0, 20}]
(* or *)
Table[((1 + I*n)*(-I + n)^n + (1 - I*n)*(I + n)^n)/2, {n, 0, 20}]

PARI

a(n) = sum(k=0, n, (n-I)^k * (n+I)^(n-k)); \\ Michel Marcus, Aug 30 2025

Formula

a(n) = ((1 + i*n)*(-i + n)^n + (1 - i*n)*(i + n)^n)/2, where i is the imaginary unit.

For n > 0, a(n) = (1 + n^2)^(n/2) * (cos(n*arctan(1/n)) + n*sin(n*arctan(1/n))).

a(n) ~ sin(1) * n^(n+1).