OFFSET
1,2
COMMENTS
Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
EXAMPLE
Triangle begins:
1
2 1
3 1 1
5 3 1 1
7 4 2 1 1
11 7 4 2 1 1
15 10 6 3 2 1 1
22 16 9 6 3 2 1 1
30 22 14 8 5 3 2 1 1
42 32 20 13 8 5 3 2 1 1
56 44 29 18 12 7 5 3 2 1 1
77 62 41 27 17 12 7 5 3 2 1 1
For example, row n = 5 counts the following partitions:
(5) (32) (32) (41) (5)
(32) (41) (311)
(41) (221)
(221) (2111)
(311)
(2111)
(11111)
At least one part appearing k or more times:
(5) (221) (2111) (11111) (11111)
(32) (311) (11111)
(41) (2111)
(221) (11111)
(311)
(2111)
(11111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], MemberQ[#/k, _?IntegerQ]&]], {n, 1, 15}, {k, 1, n}]
- or -
Table[Length[Select[IntegerPartitions[n], Max@@Length/@Split[#]>=k&]], {n, 1, 15}, {k, 1, n}]
PROG
(PARI) \\ here P(k, n) is partitions with no part divisible by k as g.f.
P(k, n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
M(n, m=n)={my(p=P(n+1, n)); Mat(vector(m, k, Col(p-P(k, n), -n) ))}
{ my(A=M(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 19 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 22 2022
STATUS
approved