W. Lang, Nov 09 2007 A134150 tabf array: partition numbers M_3(4)/M_3= M3(4)/M3. 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 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 28 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 280 28 16 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 3640 280 112 28 16 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 58240 3640 1120 784 280 112 64 28 16 4 1 0 0 0 0 0 0 0 0 0 0 0 7 1106560 58240 14560 7840 3640 1120 784 448 280 112 64 28 16 4 1 0 0 0 0 0 0 0 8 24344320 1106560 232960 101920 78400 58240 14560 7840 4480 3136 3640 1120 784 448 256 280 112 64 28 16 4 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: [608608000, 24344320, 4426240, 1630720, 1019200, 1106560, 232960, 101920, 78400, 58240, 31360, 21952, 58240, 14560, 7840, 4480, 3136, 1792, 3640, 1120, 784, 448, 256, 280, 112, 64, 28, 16, 4, 1]. n=10: [17041024000, 608608000, 97377280, 30983680, 16307200, 13249600, 24344320, 4426240, 1630720, 1019200, 931840, 407680, 313600, 219520, 1106560, 232960, 101920, 78400, 58240, 31360, 21952, 17920, 12544, 58240, 14560, 7840, 4480, 3136, 1792, 1024, 3640, 1120, 784, 448, 256, 280, 112, 64, 28, 16, 4, 1]. The first column gives A007559(n)= (3*n-2)!!! = [1,4,28,280,3640,58240,1106560,24344320,608608000,17041024000,...]. The row sums give, for n>=1: A134152(n) = [1,5,33,329,4081,64289,1193697,25959169,641756673,17842602561,...], , and coincide with those of triangle A134146. ########################################### e.o.f. #####################################################################################