Illustration of binomial transforms =================================== (by Frank.Ellermann(AT)t-online.de) If you're like me you have never heard of "binomial transforms" or even "Lambert's function". Given an unknown sequence the first thing I do is look at the successive differences: as soon as a row in the difference table contains only 0's, the sequence is presumably a polynomial in n. The N-th row of the difference table is called the "difference of 1st order and depth N" here. The 1st diagonal, i.e. the initial number in each row, is simply called the "difference of 1st order". The reverse process of writing down a sequence as the 1st diagonal, and reconstruct the difference table, is known as "binomial transform". So if sequence A is the "differences of 1st order" of sequence B, then sequence B is the "binomial transform" of sequence A. No more definitions, here's the fun: The binomial transform of the natural numbers 0,1,2,3, etc. results in sequence A001787. Skipping 0 we get A001792. Of course A001792 is the row below A001787 (aka "difference of depth 1"). The next 3 rows are A045623, A045891, and A034007. So far we found 6 known sequences, see the 1st triangle below. To get differences of 2nd order, the process is simply repeated using the differences of 1st order as input sequence. A binomial transform of A001787 gives A027471 (the 1st row of the 2nd triangle below). In other words, the differences of 2nd order of A027471 are (again) the natural numbers. Repeating this process we get 9 triangles with 1st rows corresponding to A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, and finally A053541, which is really beautiful (in base 10). The differences of 9-th order of A053541 are the natural numbers, and we found 14 = 6 + 8 known sequences ("known" in EIS). In theory this process can be repeated forever, and we find even more sequences by placing the resulting triangles one behind the other: The 2nd numbers in the 1st rows form A005843 (even numbers, boring), the 3rd numbers form A033428, and the 4th numbers are A033430. But there are more reinvented wheels: Take the 1st number in the (N+1)-st row of the N-the triangle, this is sequence A000312, the N-th power of N (N^N resp. N**N). The 1st number in the N-th row of the N-th triangle is sequence A053606, and the numbers below A053606(n) and A000312(n) are sequence A055897. +------------- A005843: (2),4,6,8,10,12,14,16,18,20,22,24,.. : +---------- A033428: (3),12,27,48,75,108,147,192,243,300,.. : : +------ A033430: (4),32,108,256,500,864,1372,2048,.. : : : A001787: 0 1 4 12 32 80 192 448 1024 2304 5120 A001792: 1 3 8 20 48 112 256 576 1280 2816 A045623: 2 5 12 28 64 144 320 704 1536 A045891: 3 7 16 36 80 176 384 832 A034007: 4 9 20 44 96 208 448 5 11 24 52 112 240 6 13 28 60 128 7 15 32 68 8 17 36 9 19 10 A027471: 0 1 6 27 108 405 1458 5103 17496 59049 196830 A053606(2): 1 5 21 81 297 1053 3645 12393 41553 137781 A000312(2): 4 16 60 216 756 2592 8748 29160 96228 A055897(3): 12 44 156 540 1836 6156 20412 67068 32 112 384 1296 4320 14256 46656 80 272 912 3024 9936 32400 192 640 2112 6912 22464 448 1472 4800 15552 1024 3328 10752 2304 7424 5120 A002697: 0 1 8 48 256 1280 6144 28672 131072 589824 2621440 1 7 40 208 1024 4864 22528 102400 458752 2031616 A053606(3); 6 33 168 816 3840 17664 79872 356352 1572864 A000312(3): 27 135 648 3024 13824 62208 276480 1216512 A055897(4): 108 513 2376 10800 48384 214272 940032 405 1863 8424 37584 165888 725760 1458 6561 29160 128304 559872 5103 22599 99144 431568 17496 76545 332424 59049 255879 196830 A053464: 0 1 10 75 500 3125 18750 109375 625000 3515625 19531250 1 9 65 425 2625 15625 90625 515625 2890625 16015625 8 56 360 2200 13000 75000 425000 2375000 13125000 A053606(4): 48 304 1840 10800 62000 350000 1950000 10750000 A000312(4): 256 1536 8960 51200 288000 1600000 8800000 A055897(5): 1280 7424 42240 236800 1312000 7200000 6144 34816 194560 1075200 5888000 28672 159744 880640 4812800 131072 720896 3932160 589824 3211264 2621440 A053469: 0 1 12 108 864 6480 46656 326592 2239488 15116544 100776960 1 11 96 756 5616 40176 279936 1912896 12877056 85660416 10 85 660 4860 34560 239760 1632960 10964160 72783360 75 575 4200 29700 205200 1393200 9331200 61819200 A053606(5): 500 3625 25500 175500 1188000 7938000 52488000 A000312(5): 3125 21875 150000 1012500 6750000 44550000 A055897(6): 18750 128125 862500 5737500 37800000 109375 734375 4875000 32062500 625000 4140625 27187500 3515625 23046875 19531250 A027473: 0 1 14 147 1372 12005 100842 823543 6588344 51883209 403536070 1 13 133 1225 10633 88837 722701 5764801 45294865 351652861 12 120 1092 9408 78204 633864 5042100 39530064 306357996 108 972 8316 68796 555660 4408236 34487964 266827932 864 7344 60480 486864 3852576 30079728 232339968 A053506(6): 6480 53136 426384 3365712 26227152 202260240 A000312(6): 46656 373248 2939328 22861440 176033088 A055897(7): 326592 2566080 19922112 153171648 2239488 17356032 133249536 15116544 115893504 100776960 A053539: 0 1 16 192 2048 20480 196608 1835008 16777216 150994944 1342177280 1 15 176 1856 18432 176128 1638400 14942208 134217728 1191182336 14 161 1680 16576 157696 1462272 13303808 119275520 1056964608 147 1519 14896 141120 1304576 11841536 105971712 937689088 1372 13377 126224 1163456 10536960 94130176 831717376 12005 112847 1037232 9373504 83593216 737587200 A053506(7): 100842 924385 8336272 74219712 653993984 A000312(7): 823543 7411887 65883440 579774272 A055897(8): 6588344 58471553 513890832 51883209 455419279 403536070 A053540: 0 1 18 243 2916 32805 354294 3720087 38263752 387420489 3874204890 1 17 225 2673 29889 321489 3365793 34543665 349156737 3486784401 16 208 2448 27216 291600 3044304 31177872 314613072 3137627664 192 2240 24768 264384 2752704 28133568 283435200 2823014592 2048 22528 239616 2488320 25380864 255301632 2539579392 20480 217088 2248704 22892544 229920768 2284277760 196608 2031616 20643840 207028224 2054356992 A053506(8): 1835008 18612224 186384384 1847328768 A000312(8): 16777216 167772160 1660944384 A055897(9): 150994944 1493172224 1342177280 A053541: 0 1 20 300 4000 50000 600000 7000000 80000000 900000000 10000000000 1 19 280 3700 46000 550000 6400000 73000000 820000000 9100000000 18 261 3420 42300 504000 5850000 66600000 747000000 8280000000 243 3159 38880 461700 5346000 60750000 680400000 7533000000 2916 35721 422820 4884300 55404000 619650000 6852600000 32805 387099 4461480 50519700 564246000 6232950000 354294 4074381 46058220 513726300 5668704000 3720087 41983839 467668080 5154977700 A053506(9): 38263752 425684241 4687309620 A000312(9): 387420489 4261625379 A055897(10): 3874204890 BTW, don't try this with Fibonacci- instead of natural numbers, all you'll get is A000045 again and again (shifted and/or signed).