W. Lang, Nov 09 2007 A134273 tabf array: partition numbers M_3(5)= M3(5). 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 45 15 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 585 180 75 30 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 9945 2925 2250 450 375 50 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 208845 59670 43875 20250 8775 13500 1875 900 1125 75 1 0 0 0 0 0 0 0 0 0 0 0 7 5221125 1461915 1044225 921375 208845 307125 141750 118125 20475 47250 13125 1575 2625 105 1 0 0 0 0 0 0 0 8 151412625 41769000 29238300 25061400 11977875 5847660 8353800 7371000 3071250 2835000 556920 1228500 567000 945000 65625 40950 126000 52500 2520 5250 140, 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: [4996616625, 1362713625, 939802500, 789434100, 733045950, 187960500, 263144700, 225552600, 107800875, 93980250, 165847500, 25515000, 17542980, 37592100, 33169500, 27641250, 25515000, 7087500, 1253070, 3685500, 1701000, 4252500, 590625, 73710, 283500, 157500, 3780, 9450, 180, 1]. n=10:[184874815125, 49966166250, 34067840625, 28194075000, 25656608250, 12461781150, 6813568125, 9398025000, 7894341000, 7330459500, 3289308750, 5638815000, 2695021875, 2487712500, 626535000, 1315723500, 1127763000, 539004375, 939802500, 1658475000, 255150000, 230343750, 318937500, 43857450, 125307000, 110565000, 138206250, 127575000, 70875000, 2953125, 2506140, 9213750, 4252500, 14175000, 2953125, 122850, 567000, 393750, 5400, 15750, 225, 1] The first column gives A007696 = [1,5,45,585,9945,208845,5221125,151412625,4996616625,184874815125,...], (4-factorials). The row sums give, for n>=1: A049120 = [1,6,61,871,15996,358891,9509641,290528316,10051973371,388433817091,...]. They coincide with the row sums of triangle A049029. ########################################### e.o.f. ############################################################################################################################