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 A183135 #42 Aug 15 2018 22:21:51
%S A183135 1,1,0,1,1,0,1,2,1,0,1,3,6,1,0,1,4,15,20,1,0,1,5,28,87,70,1,0,1,6,45,
%T A183135 232,543,252,1,0,1,7,66,485,2092,3543,924,1,0,1,8,91,876,5725,19864,
%U A183135 23823,3432,1,0,1,9,120,1435,12786,71445,195352,163719,12870,1,0
%N A183135 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0) that can be built by repeatedly inserting doublets into the initially empty word.
%C A183135 A(n,k) is also the number of rooted closed walks of length 2n on the infinite rooted k-ary tree. - _Danny Rorabaugh_, Oct 31 2017
%C A183135 A(n,2k) is the number of unreduced words of length 2n that reduce to the empty word in the free group with k generators. - _Danny Rorabaugh_, Nov 09 2017
%H A183135 Alois P. Heinz, <a href="/A183135/b183135.txt">Antidiagonals n = 0..140, flattened</a>
%H A183135 Jason Bell, Marni Mishna, <a href="https://arxiv.org/abs/1805.08118">On the Complexity of the Cogrowth Sequence</a>, arXiv:1805.08118 [math.CO], 2018.
%H A183135 Beth Bjorkman et al., <a href="https://arxiv.org/abs/1710.10616">k-foldability of words</a>, arXiv preprint arXiv:1710.10616 [math.CO], 2017.
%F A183135 A(n,k) = k/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*(k-1)^j if n>0, A(0,k) = 1.
%F A183135 A(n,k) = A183134(n,k) if n=0 or k<2, A(n,k) = A183134(n,k)*k otherwise.
%F A183135 G.f. of column k: 1/(1-k*x) if k<2, 2*(k-1)/(k-2+k*sqrt(1-(4*k-4)*x)) otherwise.
%e A183135 A(2,2) = 6, because 6 words of length 4 can be built over 2-letter alphabet {a, b} by repeatedly inserting doublets (words with two equal letters) into the initially empty word: aaaa, aabb, abba, baab, bbaa, bbbb.
%e A183135 Square array A(n,k) begins:
%e A183135 1, 1, 1, 1, 1, 1, ...
%e A183135 0, 1, 2, 3, 4, 5, ...
%e A183135 0, 1, 6, 15, 28, 45, ...
%e A183135 0, 1, 20, 87, 232, 485, ...
%e A183135 0, 1, 70, 543, 2092, 5725, ...
%e A183135 0, 1, 252, 3543, 19864, 71445, ...
%p A183135 A:= proc(n, k) local j;
%p A183135 if n=0 then 1
%p A183135 else k/n *add(binomial(2*n,j) *(n-j) *(k-1)^j, j=0..n-1)
%p A183135 fi
%p A183135 end:
%p A183135 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A183135 A[_, 1] = 1; A[n_, k_] := If[n == 0, 1, k/n*Sum[Binomial[2*n, j]*(n - j)*(k - 1)^j, {j, 0, n - 1}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A183135 Columns k=0-10 give: A000007, A000012, A000984, A089022, A035610, A130976, A130977, A130978, A130979, A130980, A131521.
%Y A183135 Rows n=0-3 give: A000012, A001477, A000384, A027849(k-1) for k>0.
%Y A183135 Main diagonal gives A294491.
%Y A183135 Coefficients of row polynomials in k, (k-1) are given by A157491, A039599.
%Y A183135 Cf. A007318, A183134, A256116, A256117.
%K A183135 nonn,tabl
%O A183135 0,8
%A A183135 _Alois P. Heinz_, Dec 26 2010