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 A213028 #24 Jan 17 2024 17:40:27
%S A213028 1,1,0,1,1,0,1,2,1,0,1,3,8,1,0,1,4,21,38,1,0,1,5,40,183,196,1,0,1,6,
%T A213028 65,508,1773,1062,1,0,1,7,96,1085,7240,18303,5948,1,0,1,8,133,1986,
%U A213028 20425,110524,197157,34120,1,0,1,9,176,3283,46476,412965,1766416,2189799,199316,1,0
%N A213028 Number A(n,k) of 3n-length k-ary words that can be built by repeatedly inserting triples of identical letters into the initially empty word; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A213028 Alois P. Heinz, <a href="/A213028/b213028.txt">Antidiagonals n = 1..140, flattened</a>
%F A213028 A(n,k) = k/n * Sum_{j=0..n-1} C(3*n,j) * (n-j) * (k-1)^j if n>0, k>1; A(0,k) = 1; A(n,k) = k if n>0, k<2.
%F A213028 A(n,k) = k * A213027(n,k) if n>0, k>1; else A(n,k) = A213027(n,k).
%e A213028 A(0,k) = 1: the empty word.
%e A213028 A(n,1) = 1: (aaa)^n.
%e A213028 A(2,2) = 8: there are 8 words of length 6 over alphabet {a,b} that can be built by repeatedly inserting triples of identical letters into the initially empty word: aaaaaa, aaabbb, aabbba, abbbaa, baaabb, bbaaab, bbbaaa, bbbbbb.
%e A213028 A(1,3) = 3: aaa, bbb, ccc.
%e A213028 A(2,3) = 21: aaaaaa, aaabbb, aaaccc, aabbba, aaccca, abbbaa, acccaa, baaabb, bbaaab, bbbaaa, bbbbbb, bbbccc, bbcccb, bcccbb, caaacc, cbbbcc, ccaaac, ccbbbc, cccaaa, cccbbb, cccccc.
%e A213028 Square array A(n,k) begins:
%e A213028 1, 1, 1, 1, 1, 1, 1, ...
%e A213028 0, 1, 2, 3, 4, 5, 6, ...
%e A213028 0, 1, 8, 21, 40, 65, 96, ...
%e A213028 0, 1, 38, 183, 508, 1085, 1986, ...
%e A213028 0, 1, 196, 1773, 7240, 20425, 46476, ...
%e A213028 0, 1, 1062, 18303, 110524, 412965, 1170066, ...
%e A213028 0, 1, 5948, 197157, 1766416, 8755985, 30921756, ...
%p A213028 A:= (n, k)-> `if`(n=0, 1,
%p A213028 k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):
%p A213028 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A213028 Unprotect[Power]; 0^0 = 1; A[n_, k_] := If[n==0, 1, k/n*Sum[Binomial[3*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 22 2017, translated from Maple *)
%Y A213028 Rows n=0-2 give: A000012, A001477, A000567.
%Y A213028 Columns k=0-2 give: A000007, A000012, A047098.
%Y A213028 Cf. A183134, A183135, A213027, A256311.
%K A213028 nonn,tabl
%O A213028 0,8
%A A213028 _Alois P. Heinz_, Jun 03 2012