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A354234
Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.
4
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
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
The complement is counted by A061199.
Differences of consecutive terms are A091602.
Column k = 1 is A000041.
Column k = 2 is A047967, ranked by A013929 and A324929.
Column k = 3 is A295341, ranked by A046099 and A354235.
Column k = 4 is A295342.
A000041 counts integer partitions, strict A000009.
A047966 counts uniform partitions.
Sequence in context: A106740 A178534 A110619 * A191861 A350200 A129761
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 22 2022
STATUS
approved