from fractions import Fraction
import decimal
import math
from math import factorial


def calc_binomial(a: int, b: int) -> int:
    a_fact = factorial(a)
    b_fact = factorial(b)
    a_b_fact = factorial(a - b)
    denominator = b_fact * a_b_fact

    # This has to be // for some reason...
    result = a_fact // denominator
    assert result == math.comb(a, b)

    return result


def count_sequences(n: int):
    if n % 2 != 0:
        return 0

    n_div = n // 2
    assert n_div.is_integer()

    binomial = calc_binomial(n, n_div)
    assert binomial.is_integer()

    # oh man this took time...
    result_div = Fraction(1, (n_div + 1))
    result = round(result_div * binomial)

    return int(result)


if __name__ == "__main__":
    assert count_sequences(1) == 0
    assert count_sequences(2) == 1
    assert count_sequences(3) == 0
    assert count_sequences(4) == 2
    assert count_sequences(5) == 0
    assert count_sequences(6) == 5

    assert count_sequences(42) == 24466267020
    assert count_sequences(62) == 14544636039226909
    assert count_sequences(64) == 55534064877048198
    assert (
        count_sequences(1000)
        == 539497486917039060909410566119711128734834348196703167679426896420410037336371644508208550747509720888947317534973145917768881736628103627844100238921194561723883202123256952806711505149177419849031086149939116975191706558395784192643914160118616272189452807591091542120727401415762287153293056320
    )
    assert count_sequences(5000)