cp's OEIS Frontend

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.

%I A366154 #13 Oct 09 2023 09:24:39
%S A366154 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 A366154 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 A366154 7,2,176,112,61,39,23,12,4,1,231,142,81,52,32,18,7,2
%N 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.
%C A366154 Empirical: The first k terms of each column are A000070, for columns k > 0.
%H A366154 John Tyler Rascoe, <a href="/A366154/a366154.py.txt">Python program</a>
%e A366154 Triangle begins:
%e A366154       k=0   1  2  3  4
%e A366154   n=0:  0
%e A366154   n=1:  1
%e A366154   n=2:  2
%e A366154   n=3:  3,  1
%e A366154   n=4:  5,  1
%e A366154   n=5:  7,  3, 1
%e A366154   n=6: 11,  3, 2
%e A366154   n=7: 15,  7, 3, 1
%e A366154   n=8: 22,  9, 4, 2
%e A366154   n=9: 30, 15, 7, 4, 1
%e A366154   ...
%e A366154 T(7,1) = 7: T(7,2) = 3: T(7,3) = 1:
%e A366154       (43)        (331)      (4111)
%e A366154      (421)       (3211)
%e A366154      (322)      (31111)
%e A366154     (3211)
%e A366154     (2221)
%e A366154    (22111)
%e A366154   (211111)
%o A366154 (Python) # see linked program
%Y A366154 Cf. A000041 (column k=0), A237666 (column k=1).
%Y A366154 Cf. A000070, A008615, A268193, A354234.
%K A366154 nonn,tabf
%O A366154 0,3
%A A366154 _John Tyler Rascoe_, Oct 01 2023