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

0,2

Links

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

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).

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}]

Prog

(PARI) a(n) = sum(k=0, n, (n-I)^k * (n+I)^(n-k)); \\ Michel Marcus, Aug 30 2025
(Magma) C<I> := 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

Crossrefs

Cf. A387430.

Sequence in context: A261886 A245895 A231229 * A138210 A227465 A295099

Adjacent sequences: A387456 A387457 A387458 * A387460 A387461 A387462

Keyword

nonn

Author

Vaclav Kotesovec, Aug 29 2025

Status

approved