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 A299807 #34 Aug 15 2022 15:31:06 %S A299807 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,9,10,5,1,1,6,15,16,15,6,1,1,7,19, %T A299807 35,25,21,7,1,1,8,28,37,70,36,28,8,1,1,9,33,84,61,126,49,36,9,1,1,10, %U A299807 45,96,210,91,210,64,45,10,1,1,11,51,163,225,462,127,330,81,55,11,1,1,12,66,180,477,456,924,169,495,100,66 %N A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0. %H A299807 Max Alekseyev, <a href="/A299807/b299807.txt">Table of n, a(n) for n = 1..351</a> %F A299807 From _Chai Wah Wu_, May 28 2018: (Start) %F A299807 The following are all conjectures. %F A299807 For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1). %F A299807 For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p. %F A299807 (End) %e A299807 Array starts: %e A299807 n=1: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A299807 n=2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... %e A299807 n=3: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ... %e A299807 n=4: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ... %e A299807 n=5: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, ... %e A299807 n=6: 1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, ... %e A299807 n=7: 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, ... %e A299807 n=8: 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, ... %e A299807 n=9: 1, 9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ... %e A299807 n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951, 6531, 10201, 15231, ... %e A299807 ... %Y A299807 Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8). %Y A299807 Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2). %Y A299807 Diagonal: A299754 (n=k). %Y A299807 Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008. %K A299807 nonn,tabl %O A299807 1,5 %A A299807 _Max Alekseyev_, Feb 24 2018 %E A299807 Row 6 corrected by _Max Alekseyev_, Aug 14 2022