OFFSET
1,31
COMMENTS
Row sums = A027293.
If superfluous zeros are removed from the right side of each row, the row lengths = 1,2,1,3,1,1,4,2,... = A010766.
Sum of each N X N block of rows = 1,2,4,7,12,19,... = A000070.
The sum of the partitions of n that are over-counted in each block of N x N rows = A000070(n) - A000041(n) = A058884(n), n >= 1.
Concatenation of first row from each N X N block = A116598.
As noted by Joerg Arndt in A116598, the first row from each N X N block in reverse converges to A002865. Two sequences emerge from alternating second rows in reverse: for 2n, converges to even-indexed terms in A027336, and for 2n+1, converges to odd-indexed terms in A027336.
Counting the rows in each N X N block where columns j=2 > 0 and j=3 through j=n are all zeros produces A008615(n), n > 0.
LINKS
J. Stauduhar, Table of n, a(n) for n = 1..10000
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014.
J. Stauduhar, Original Python program.
EXAMPLE
\ j 1 2 3 4 5
k
n
1: 1 1
2: 1 0 1
2 1 0
3: 1 1 0 1
2 1 0 0
3 1 0 0
4: 1 1 1 0 1
2 1 1 0 0
3 1 0 0 0
4 1 0 0 0
5: 1 2 1 1 0 1
2 2 1 0 0 0
3 2 0 0 0 0
4 1 0 0 0 0
5 1 0 0 0 0
.
.
.
MATHEMATICA
Array[With[{s = IntegerPartitions[#]}, Table[Count[Map[Count[#, k] &, s], j], {k, #}, {j, #}]] &, 7] // Flatten (* Michael De Vlieger, Feb 28 2018 *)
PROG
(PYTHON) See Stauduhar link.
CROSSREFS
KEYWORD
nonn,tabf,changed
AUTHOR
J. Stauduhar, Feb 27 2018
STATUS
approved