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.

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

Author

John Tyler Rascoe, Oct 01 2023

Keywords

• nonn
• tabf

Comments

Empirical: The first k terms of each column are A000070, for columns k > 0.

Examples

			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)
		

Links

• John Tyler Rascoe, Python program

Crossrefs

Cf. A000041 (column k=0), A237666 (column k=1).

Cf. A000070, A008615, A268193, A354234.

Programs

Python

# see linked program