W. Lang, Sep 19 2008 A107106 tabf array: partition numbers M31hat(1) (W.L. notation) is M_2/M_3 (A-St notation). Row n is filled with zeros for k>p(n), the partition number. Partitions of n listed in Abramowitz-Stegun (A-St) 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 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 24 6 2 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 120 24 6 4 6 2 1 2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 7 720 120 24 12 24 6 4 2 6 2 1 2 1 1 1 0 0 0 0 0 0 0 8 5040 720 120 48 36 120 24 12 6 4 24 6 4 2 1 6 2 1 2 1 1 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 .. n=9: [40320, 5040, 720, 240, 144, 720, 120, 48, 36, 24, 12, 8, 120, 24, 12, 6, 4, 2, 24, 6, 4, 2, 1, 6, 2, 1, 2, 1, 1, 1], n=10: [362880, 40320, 5040, 1440, 720, 576, 5040, 720, 240, 144, 120, 48, 36, 24, 720, 120, 48, 36, 24, 12, 8, 6, 4, 120, 24, 12, 6, 4, 2, 1, 24, 6, 4, 2, 1, 6, 2, 1, 2, 1, 1, 1]. The first column gives (n-1)!= A000142(n-1), n>=1, (factorials). The row sums give for n>=1: A107107. They coincide with the row sums of triangle A144351. ########################################### e.o.f. ############################################################################################################################