A235130 Triangular array: t(n,k) = number of partitions of n that include a partition of k.

1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 8, 6, 7, 7, 11, 11, 11, 11, 11, 10, 11, 15, 15, 17, 15, 14, 13, 15, 15, 22, 22, 23, 23, 21, 22, 19, 20, 22, 30, 30, 33, 30, 33, 25, 29, 25, 29, 30, 42, 42, 45, 44, 43, 41, 42, 36, 36, 39, 42, 56, 56, 62, 58, 60

Offset: 1

Author

Clark Kimberling, Jan 03 2014

Keywords

• nonn
• tabl

Examples

			The eleven partitions of 6 include the following six, written as multisets:  {1,1,1,1,1,1}, {1,1,1,1,2}, {1,1,2,2}, {1,1,1,3}, {1,2,3}, {3,3}; each has a sub-multiset of which the sum of terms is 3.  None of the remaining five partitions of 6 has this property, so t(6,3) = 6.  First 7 rows:
1
1 ... 2
2 ... 2 ... 3
3 ... 3 ... 3 ... 5
5 ... 5 ... 5 ... 5 ... 7
7 ... 8 ... 6 ... 7 ... 7 ... 11
11 .. 11 .. 11 .. 11 .. 10 .. 11 .. 15
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A000041.

Programs

Mathematica

p[n_] := p[n] = IntegerPartitions[n]; t = Table[Length[Cases[p[n], Apply[Alternatives, Map[Flatten[{_, #, _}] &, p[k]]]]], {n, 15}, {k, n}]; u = Flatten[t] (* 235130 *)
TableForm[t] (* Peter J. C. Moses, Jan 04 2014 *)

Formula

t(n,1) = A000041(n-1) for n>=0; t(n,n) = A000041(n) for n >= 1.