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 A260877 #28 Mar 06 2020 03:12:58
%S A260877 1,1,-1,1,-1,1,1,-1,1,-5,1,-1,5,-1,21,1,-1,19,-61,1,-105,1,-1,69,
%T A260877 -1513,1385,-1,635,1,-1,251,-33661,315523,-50521,1,-4507,1,-1,923,
%U A260877 -750751,60376809,-136085041,2702765,-1,36457,1,-1,3431,-17116009,11593285251
%N A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers.
%C A260877 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. It is ordered if the positions of the blocks are taken into account.
%C A260877 M-shape Euler numbers count the ordered m-shape set partitions which have even length minus the number of such partitions which have odd length.
%C A260877 If m=0 all possible sizes are zero. Thus m-shape Euler numbers count the ordered integer partitions of n into an even number of parts minus the number of ordered integer partitions of n into an odd number of parts (A260845).
%C A260877 If m=1 the set is {1,2,...,n} and the set of all possible sizes are the integer partitions of n. Thus the Euler numbers count the ordered set partitions which have even length minus the set partitions which have odd length (A033999).
%C A260877 If m=2 the set is {1,2,...,2n} and the 2-shape Euler numbers count the ordered set partitions with even blocks which have even length minus the number of partitions with even blocks which have odd length (A028296).
%e A260877 [ n ] [0 1 2 3 4 5 6]
%e A260877 [ m ] --------------------------------------------------------------
%e A260877 [ 0 ] [1, -1, 1, -5, 21, -105, 635] A260845
%e A260877 [ 1 ] [1, -1, 1, -1, 1, -1, 1] A033999
%e A260877 [ 2 ] [1, -1, 5, -61, 1385, -50521, 2702765] A028296
%e A260877 [ 3 ] [1, -1, 19, -1513, 315523, -136085041, 105261234643] A002115
%e A260877 [ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212
%e A260877 A030662,A211213, A181991,
%e A260877 For example the number of ordered set partitions of {1,2,...,9} with sizes in [9], [6,3] and [3,3,3] are 1, 168, 1680 respectively. Thus A(3,3) = -1 + 168 - 1680 = -1513.
%e A260877 Formatted as a triangle:
%e A260877 [1]
%e A260877 [1, -1]
%e A260877 [1, -1, 1]
%e A260877 [1, -1, 1, -5]
%e A260877 [1, -1, 5, -1, 21]
%e A260877 [1, -1, 19, -61, 1, -105]
%e A260877 [1, -1, 69, -1513, 1385, -1, 635]
%o A260877 (Sage)
%o A260877 def A260877(m,n):
%o A260877 shapes = ([x*m for x in p] for p in Partitions(n).list())
%o A260877 return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes)
%o A260877 for m in (0..5): print([A260877(m,n) for n in (0..7)])
%Y A260877 Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876.
%K A260877 sign,tabl
%O A260877 1,10
%A A260877 _Peter Luschny_, Aug 09 2015