W. Lang Oct 09 2008 A144890 tabf array: partition numbers M31hat(4). 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 30 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 210 30 25 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1680 210 150 30 25 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 15120 1680 1050 900 210 150 125 30 25 5 1 0 0 0 0 0 0 0 0 0 0 0 7 151200 15120 8400 6300 1680 1050 900 750 210 150 125 30 25 5 1 0 0 0 0 0 0 0 8 1663200 151200 75600 50400 44100 15120 8400 6300 5250 4500 1680 1050 900 750 625 210 150 125 30 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 next two rows, for n=9 and n=10, are: n=9: [19958400, 1663200, 756000, 453600, 352800, 151200, 75600, 50400, 44100, 42000, 31500, 27000, 15120, 8400, 6300, 5250, 4500, 3750, 1680, 1050, 900, 750, 625, 210, 150, 125, 30, 25, 5, 1], n=10: [259459200, 19958400, 8316000, 4536000, 3175200, 2822400, 1663200, 756000, 453600, 352800, 378000, 52000, 220500, 189000, 151200, 75600, 50400, 44100, 42000, 31500, 27000, 26250, 22500, 15120, 8400, 6300, 5250, 4500, 3750, 3125, 1680, 1050, 900, 750, 625, 210, 150, 125, 30, 25, 5, 1]. The row sums give, for n>=1: A144892 = [1,6,36,271,2101,19296,185946,2029621,23654671,303054846,...]. They coincide with the row sums of triangle A144891 = S1hat(5). ########################################### e.o.f. #####################################################################################