A317659 Regular triangle where T(n,k) is the number of distinct free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and k leaves.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 43, 33, 9, 1, 0, 1, 21, 92, 118, 55, 11, 1, 0, 1, 28, 174, 341, 252, 82, 13, 1, 0, 1, 36, 302, 845, 935, 463, 115, 15, 1, 0, 1, 45, 490, 1864, 2921, 2103, 769, 153, 17, 1, 0, 1, 55, 755

Offset: 1

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn
• tabl

Examples

			The T(5,3) = 5 expressions are o[o[o]], o[o,o[]], o[][o,o], o[o][o], o[o,o][].
Triangle begins:
    1
    1    0
    1    1    0
    1    3    1    0
    1    6    5    1    0
    1   10   17    7    1    0
    1   15   43   33    9    1    0
    1   21   92  118   55   11    1    0
    1   28  174  341  252   82   13    1    0
    1   36  302  845  935  463  115   15    1    0
    1   45  490 1864 2921 2103  769  153   17    1    0
    1   55  755 3755 7981 8012 4145 1187  197   19    1    0
		

Links

• Mathematica Reference, Orderless

Crossrefs

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317654, A317655, A317656, A317676.

Programs

Mathematica

maxUsing[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h],Union[Sort/@Tuples[maxUsing/@p]]}],{p,IntegerPartitions[g]}]]];
Table[Length[Select[maxUsing[n],Length[Position[#,"o"]]==k&]],{n,12},{k,n}]