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A366154
Irregular triangle read by rows: T(n,k) is the number of integer partitions of n with at least one part a_i such that a_i - a_{i+k} = k.
1
0, 1, 2, 3, 1, 5, 1, 7, 3, 1, 11, 3, 2, 15, 7, 3, 1, 22, 9, 4, 2, 30, 15, 7, 4, 1, 42, 20, 11, 6, 2, 56, 32, 16, 9, 4, 1, 77, 40, 22, 12, 7, 2, 101, 61, 33, 19, 11, 4, 1, 135, 78, 44, 26, 16, 7, 2, 176, 112, 61, 39, 23, 12, 4, 1, 231, 142, 81, 52, 32, 18, 7, 2
OFFSET
0,3
COMMENTS
Empirical: The first k terms of each column are A000070, for columns k > 0.
EXAMPLE
Triangle begins:
k=0 1 2 3 4
n=0: 0
n=1: 1
n=2: 2
n=3: 3, 1
n=4: 5, 1
n=5: 7, 3, 1
n=6: 11, 3, 2
n=7: 15, 7, 3, 1
n=8: 22, 9, 4, 2
n=9: 30, 15, 7, 4, 1
...
T(7,1) = 7: T(7,2) = 3: T(7,3) = 1:
(43) (331) (4111)
(421) (3211)
(322) (31111)
(3211)
(2221)
(22111)
(211111)
PROG
(Python) # see linked program
CROSSREFS
Cf. A000041 (column k=0), A237666 (column k=1).
Sequence in context: A166333 A322966 A173239 * A214055 A066909 A095195
KEYWORD
nonn,tabf
AUTHOR
John Tyler Rascoe, Oct 01 2023
STATUS
approved