This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A128627 #12 Oct 03 2022 08:46:46 %S A128627 1,0,1,1,0,1,1,2,0,1,2,2,3,0,1,2,5,3,4,0,1,4,6,9,4,5,0,1,4,13,12,14,5, %T A128627 6,0,1,7,16,28,20,20,6,7,0,1,8,30,39,50,30,27,7,8,0,1,12,40,78,76,80, %U A128627 42,35,8,9,0,1,14,66,115,161,130,119,56,44,9,10,0,1 %N A128627 Triangle read by rows. Convolution triangle based on A002865. %C A128627 Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds (cf. A060850). %C A128627 The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences). %C A128627 4 1 2 3 %C A128627 22 1 3 6 %C A128627 4' 2 5 9 %C A128627 5 1 2 3 4 %C A128627 32 1 4 9 16 %C A128627 5' 2 6 12 20 %C A128627 6 1 2 3 4 5 6 7 8 9 %C A128627 42 1 4 9 16 25 36 49 64 81 %C A128627 33 1 3 6 10 15 21 28 36 45 %C A128627 222 1 4 10 20 35 56 84 120 165 %C A128627 6' 4 13 28 50 80 119 168 228 300 %C A128627 7 1 2 3 4 5 6 7 8 9 %C A128627 52 1 4 9 16 25 36 49 64 81 %C A128627 43 1 4 9 16 25 36 49 64 81 %C A128627 322 1 6 18 40 75 126 196 288 405 %C A128627 7' 4 16 39 76 130 204 301 424 576 %C A128627 8 1 2 3 4 5 6 7 8 9 %C A128627 62 1 4 9 16 25 36 49 64 81 %C A128627 53 1 4 9 16 25 36 49 64 81 %C A128627 44 1 3 6 10 15 21 28 36 45 %C A128627 422 1 6 18 40 75 126 196 288 405 %C A128627 332 1 6 18 40 75 126 196 288 405 %C A128627 2222 1 5 15 35 70 126 210 330 495 %C A128627 8' 7 30 78 161 290 477 735 1078 1521 %e A128627 The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated: %e A128627 1 %e A128627 0 1 %e A128627 1 0 1 %e A128627 1 2 0 1 %e A128627 2 2 3 0 %e A128627 2 5 3 4 %e A128627 4 6 9 4 %e A128627 4 13 12 14 %e A128627 7 16 28 20 %e A128627 30 39 50 %e A128627 78 76 %e A128627 161 %e A128627 times %e A128627 1 %e A128627 1 9 %e A128627 1 8 45 %e A128627 1 7 36 165 %e A128627 1 6 28 120 %e A128627 1 5 21 84 %e A128627 1 4 15 56 %e A128627 1 3 10 35 %e A128627 1 2 6 20 %e A128627 1 3 10 %e A128627 1 4 %e A128627 1 %e A128627 yields %e A128627 1 %e A128627 0 9 %e A128627 1 0 45 %e A128627 1 14 0 165 %e A128627 2 12 84 0 %e A128627 2 25 63 336 %e A128627 4 24 135 224 %e A128627 4 39 120 490 %e A128627 7 32 168 400 %e A128627 30 117 500 %e A128627 78 304 %e A128627 161 %e A128627 summing to %e A128627 22 185 810 2580 ... %e A128627 Triangle T(n, k) starts: %e A128627 [ 1] 1; %e A128627 [ 2] 0, 1; %e A128627 [ 3] 1, 0, 1; %e A128627 [ 4] 1, 2, 0, 1; %e A128627 [ 5] 2, 2, 3, 0, 1; %e A128627 [ 6] 2, 5, 3, 4, 0, 1; %e A128627 [ 7] 4, 6, 9, 4, 5, 0, 1; %e A128627 [ 8] 4, 13, 12, 14, 5, 6, 0, 1; %e A128627 [ 9] 7, 16, 28, 20, 20, 6, 7, 0, 1; %e A128627 [10] 8, 30, 39, 50, 30, 27, 7, 8, 0, 1; %p A128627 # Using function A002865 and function PMatrix from A357368. %p A128627 A128627Triangle := proc(dim) local M, Row, r; %p A128627 M := PMatrix(dim, n -> A002865(n-1)); %p A128627 Row := r -> convert(linalg:-row(M, r), list)[2..r]; %p A128627 for r from 2 to dim do lprint(Row(r)) od end: %p A128627 A128627Triangle(11); # _Peter Luschny_, Oct 03 2022 %Y A128627 Cf. A002865, A060850, A007318. %K A128627 nonn,tabl %O A128627 1,8 %A A128627 _Alford Arnold_, Mar 22 2007 %E A128627 New name by _Peter Luschny_, Oct 03 2022