Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%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