Is it permissible for a father to record the amounts paid to him by one of his children as a debt owed by him to this child, to be deducted from the share of the other child in the inheritance after the father's death, especially if the other child would then be left with nothing from the inheritance or become indebted to his brother?

The ruling on a son helping his father with money depends on the son's intention. If he intends it as an act of piety (birr) and a gift, then he has no right to demand it back, and he will be rewarded for it. If he intends it as a loan, then he has the right to demand it back during his father's lifetime or from his estate after his death, and what the father wrote may serve as evidence for this. However, if the father considers what the son paid him as a debt upon himself, without the son intending it as a loan, then this does not entitle the son to a right over his father if the son was a volunteer; because a loan requires an offer and acceptance, and the mere acceptance of the father is not sufficient without the son's offer.