W. Lang, Sep 18 2008 A144341 tabf array: partition numbers M32hat(-5)= 'M32(-5)/M3'. Row n is filled with zeros for k>p(n), the partition number. Partitions of n listed in Abramowitz-Stegun order p. 831-2 (see the main page for an A-number with the reference). n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 55 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 935 55 25 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 21505 935 275 55 25 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 623645 21505 4675 3025 935 275 125 55 25 5 1 0 0 0 0 0 0 0 0 0 0 0 7 21827575 623645 107525 51425 21505 4675 3025 1375 935 275 125 55 25 5 1 0 0 0 0 0 0 0 8 894930575 21827575 3118225 1182775 874225 623645 107525 51425 23375 15125 21505 4675 3025 1375 625 935 275 125 55 25 5 1 . . . n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 .. The rows n=9 and n=10 are: n=9: [42061737025, 894930575, 109137875, 34300475, 20107175, 21827575, 3118225, 1182775, 874225, 537625, 257125, 166375, 623645, 107525, 51425, 23375, 15125, 6875, 21505, 4675, 3025, 1375, 625, 935, 275, 125, 55, 25, 5, 1]. n=10: [2229272062325, 42061737025, 4474652875, 1200516625, 583108075, 462465025, 894930575, 109137875, 34300475, 20107175, 15591125, 5913875, 4371125, 2828375, 21827575, 3118225, 1182775, 874225, 537625, 257125, 166375, 116875, 75625, 623645, 107525, 51425, 23375, 15125, 6875, 3125, 21505, 4675, 3025, 1375, 625, 935, 275, 125, 55, 25, 5, 1]. The first column gives A008543(n-1)=(6*n-7)(!^7),n>=2, (6-factorials) and 1 for n=1. The row sums give for n>=1: A144343= [1,6,61,1021,22801,654271,22642171,922787096,43149037646,2279170742696,...]. They coincide with the row sums of triangle S2hat(-5)= A144342. ########################################### e.o.f. ############################################################################################################################