N.H. Rev. Stat. Ann. § 166:19

Any county furnishing any assistance to any person within 6 years preceding his death shall be entitled to recover from the estate of such person the sum or sums paid out for such assistance, the same to be a preferred claim against his said estate after the payment of his funeral charges, expense of last sickness and expenses of administration, provided he leaves no widow or minor children living at his decease.