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

1,3

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A000041.

Sequence in context: A157873 A022870 A237050 * A131410 A202453 A259529

Adjacent sequences: A235127 A235128 A235129 * A235131 A235132 A235133

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 03 2014

Status

approved