

A235130


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


4



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
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OFFSET

1,3


LINKS

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


FORMULA

t(n,1) = A000041(n1) 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 submultiset 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



