This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A260875 #27 Mar 16 2020 08:51:35
%S A260875 1,1,-1,1,-1,0,1,-1,0,-1,1,-1,2,1,1,1,-1,9,-1,1,-1,1,-1,34,-197,-43,
%T A260875 -2,1,1,-1,125,-5281,6841,254,-9,-1,1,-1,461,-123124,2185429,-254801,
%U A260875 4157,-9,2,1,-1,1715,-2840293,465693001,-1854147586,-3000807,-70981,50,-2
%N A260875 Square array read by ascending antidiagonals: number of m-shape complementary Bell numbers.
%C A260875 A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n.
%C A260875 M-complementary Bell numbers count the m-shape set partitions which have even length minus the number of such partitions which have odd length.
%C A260875 If m=0 all possible sizes are zero. Thus in this case the complementary Bell numbers count the integer partitions of n into an even number of parts minus the number of integer partitions of n into an odd number of parts (A081362).
%C A260875 If m=1 the set is {1,2,...,n} and the complementary Bell numbers count the set partitions which have even length minus the set partitions which have odd length (A000587).
%C A260875 If m=2 the set is {1,2,...,2n} and the complementary Bell numbers count the set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A260884).
%e A260875 [ n ] [ 0 1 2 3 4 5 6]
%e A260875 [ m ] --------------------------------------------------------
%e A260875 [ 0 ] [ 1, -1, 0, -1, 1, -1, 1] A081362
%e A260875 [ 1 ] [ 1, -1, 0, 1, 1, -2, -9] A000587
%e A260875 [ 2 ] [ 1, -1, 2, -1, -43, 254, 4157] A260884
%e A260875 [ 3 ] [ 1, -1, 9, -197, 6841, -254801, -3000807]
%e A260875 [ 4 ] [ 1, -1, 34, -5281, 2185429, -1854147586, 2755045819549]
%e A260875 A010763,
%e A260875 For example the number of set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 84, 280 respectively. Thus A(3,3) = -1 + 84 - 280 = -197.
%e A260875 Formatted as a triangle:
%e A260875 [1]
%e A260875 [1, -1]
%e A260875 [1, -1, 0]
%e A260875 [1, -1, 0, -1]
%e A260875 [1, -1, 2, 1, 1]
%e A260875 [1, -1, 9, -1, 1, -1]
%e A260875 [1, -1, 34, -197, -43, -2, 1]
%e A260875 [1, -1, 125, -5281, 6841, 254, -9, -1]
%o A260875 (Sage)
%o A260875 def A260875(m, n):
%o A260875 shapes = ([x*m for x in p] for p in Partitions(n))
%o A260875 return sum((-1)^len(s)*SetPartitions(sum(s),s).cardinality() for s in shapes)
%o A260875 for m in (0..4): print([A260875(m,n) for n in (0..6)])
%Y A260875 Cf. A000587, A010763, A081362, A260877, A260833, A260884, A260876.
%K A260875 sign,easy,tabl
%O A260875 1,13
%A A260875 _Peter Luschny_, Aug 09 2015