W. Lang, Sep 18 2008 A144279 tabf array: partition numbers M32hat(-3)= 'M32(-3)/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 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 21 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 231 21 9 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 3465 231 63 21 9 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 65835 3465 693 441 231 63 27 21 9 3 1 0 0 0 0 0 0 0 0 0 0 0 7 1514205 65835 10395 4851 3465 693 441 189 231 63 27 21 9 3 1 0 0 0 0 0 0 0 8 40883535 1514205 197505 72765 53361 65835 10395 4851 2079 1323 3465 693 441 189 81 231 63 27 21 9 3 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: [1267389585, 40883535, 4542615, 1382535, 800415, 1514205, 197505, 72765, 53361, 31185, 14553, 9261, 65835, 10395, 4851, 2079, 1323, 567, 3465, 693, 441, 189, 81, 231, 63, 27, 21, 9, 3, 1], n=10:[44358635475, 1267389585, 122650605, 31798305, 15207885, 12006225, 40883535, 4542615, 1382535, 800415, 592515, 218295, 160083, 101871, 1514205, 197505, 72765, 53361, 31185, 14553, 9261, 6237, 3969, 65835, 10395, 4851, 2079, 1323, 567, 243, 3465, 693, 441, 189, 81, 231, 63, 27, 21, 9, 3, 1]. The first column gives A008545(n-1)=(4*n-5)(!^4),n>=2, (4-factorials) and 1 for n=1. The row sums give for n>=1: A144281= [1,4,25,265,3793,70789,1600429,42811078,1316981794,45858363502,...]. They coincide with the row sums of triangle S2hat(-3)= A144280. ########################################### e.o.f. ############################################################################################################################