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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

%I #13 Oct 09 2023 09:24:39

%S 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,

%T 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,

%U 7,2,176,112,61,39,23,12,4,1,231,142,81,52,32,18,7,2

%N 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.

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

%H John Tyler Rascoe, <a href="/A366154/a366154.py.txt">Python program</a>

%e Triangle begins:

%e k=0 1 2 3 4

%e n=0: 0

%e n=1: 1

%e n=2: 2

%e n=3: 3, 1

%e n=4: 5, 1

%e n=5: 7, 3, 1

%e n=6: 11, 3, 2

%e n=7: 15, 7, 3, 1

%e n=8: 22, 9, 4, 2

%e n=9: 30, 15, 7, 4, 1

%e ...

%e T(7,1) = 7: T(7,2) = 3: T(7,3) = 1:

%e (43) (331) (4111)

%e (421) (3211)

%e (322) (31111)

%e (3211)

%e (2221)

%e (22111)

%e (211111)

%o (Python) # see linked program

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

%Y Cf. A000070, A008615, A268193, A354234.

%K nonn,tabf

%O 0,3

%A _John Tyler Rascoe_, Oct 01 2023