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 A260877 Square array read by ascending antidiagonals: number of m-shape Euler numbers. 2
 1, 1, -1, 1, -1, 1, 1, -1, 1, -5, 1, -1, 5, -1, 21, 1, -1, 19, -61, 1, -105, 1, -1, 69, -1513, 1385, -1, 635, 1, -1, 251, -33661, 315523, -50521, 1, -4507, 1, -1, 923, -750751, 60376809, -136085041, 2702765, -1, 36457, 1, -1, 3431, -17116009, 11593285251 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS 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. 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. 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). 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). 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). LINKS EXAMPLE [ n ] [0   1   2       3         4              5                 6] [ m ] -------------------------------------------------------------- [ 0 ] [1, -1,  1,     -5,       21,          -105,              635] A260845 [ 1 ] [1, -1,  1,     -1,        1,            -1,                1] A033999 [ 2 ] [1, -1,  5,    -61,     1385,        -50521,          2702765] A028296 [ 3 ] [1, -1, 19,  -1513,   315523,    -136085041,     105261234643] A002115 [ 4 ] [1, -1, 69, -33661, 60376809, -288294050521, 3019098162602349] A211212 For example the number of ordered set partitions of {1,2,...,9} with sizes in , [6,3] and [3,3,3] are 1, 168, 1680 respectively. Thus A(3,3) = -1 + 168 - 1680 = -1513. Formatted as a triangle:  [1, -1] [1, -1,  1] [1, -1,  1,    -5] [1, -1,  5,    -1,   21] [1, -1, 19,   -61,    1, -105] [1, -1, 69, -1513, 1385,   -1, 635] PROG (Sage) def A260877(m, n):     shapes = ([x*m for x in p] for p in Partitions(n).list())     return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s). cardinality() for s in shapes) for m in (0..5): print([A260877(m, n) for n in (0..7)]) CROSSREFS Cf. A002115, A028296, A030662, A033999, A181991, A211212, A211213, A260845, A260833, A260875, A260876. Sequence in context: A010333 A131777 A323388 * A237888 A286462 A046607 Adjacent sequences:  A260874 A260875 A260876 * A260878 A260879 A260880 KEYWORD sign,tabl AUTHOR Peter Luschny, Aug 09 2015 STATUS approved

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Last modified October 25 20:21 EDT 2021. Contains 348256 sequences. (Running on oeis4.)