Irregular triangle of numbers m for which binomial(S001077(n),m) is squarefree.

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 16, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

1

Note that the squarefree values in binomial(n,m) occur for small m when n is large.

T. D. Noe, Plot of 16 rows of terms

T. D. Noe, Table of 16 rows of terms

Wikipedia, Binomial coefficient

(Mma) mx = -1; t = {}; Do[temp = Length[s = Select[Range[0, n/2], SquareFreeQ[Binomial[n, #]] &]]; If[temp > mx, mx = temp; AppendTo[t, {n, s}]], {n, 0, 1000}]; Flatten[Transpose[t][[2]]]

Cf. A238337, S001077, S001078.

nonn,tabf

T. D. Noe, Jan 25 2018