A236972 The number of partitions of n into at least 5 parts from which we can form every partition of n into 5 parts by summing elements.

0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 7, 8, 9, 13, 14, 24, 29, 35, 38

Offset

1,6

Comments

The corresponding partitions with 2 in the definition instead of 5 are the complete partitions, which A126796 counts.

The qualifier 'into at least 5 parts' is only relevant for n = 1, 2, 3 or 4. It is included because otherwise the condition would be vacuously true for all partitions of 1, 2, 3 and 4. It seems neater to consider that there are no partitions of 1, 2, 3 or 4 of this form.

What is the limit for large n of the proportion of partitions of n for which this holds, or this sequence divided by A000041?

Links

Table of n, a(n) for n=1..19.

Example

The valid partitions of 11 are all those which contain only 1's, 2's and 3's, with no more than one 3 and no more than three 2's or 3's. This is because every partition of 11 into 5 parts contains at least one element 3 or more, and at least 3 elements 2 or more. There are 7 such partitions, therefore a(11) = 7.

Crossrefs

A000041 counts partitions, A126796 counts complete partitions - the case for partitions into 2 instead of 5, A236970 and A236971 are the cases for 3 and 4 respectively.

Sequence in context: A164529 A153906 A067619 * A184351 A343659 A146922

Adjacent sequences: A236969 A236970 A236971 * A236973 A236974 A236975

Keyword

nonn,more

Author

Jack W Grahl, Feb 02 2014

Status

approved