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.

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, 232, 543, 252, 1, 0, 1, 7, 66, 485, 2092, 3543, 924, 1, 0, 1, 8, 91, 876, 5725, 19864, 23823, 3432, 1, 0, 1, 9, 120, 1435, 12786, 71445, 195352, 163719, 12870, 1, 0

Offset: 0

Author

Alois P. Heinz, Dec 26 2010

Keywords

• nonn
• tabl

Comments

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

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

Examples

			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.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,  ...
  0,  1,   2,    3,     4,     5,  ...
  0,  1,   6,   15,    28,    45,  ...
  0,  1,  20,   87,   232,   485,  ...
  0,  1,  70,  543,  2092,  5725,  ...
  0,  1, 252, 3543, 19864, 71445,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Jason Bell, Marni Mishna, On the Complexity of the Cogrowth Sequence, arXiv:1805.08118 [math.CO], 2018.
• Beth Bjorkman et al., k-foldability of words, arXiv preprint arXiv:1710.10616 [math.CO], 2017.

Crossrefs

Columns k=0-10 give: A000007, A000012, A000984, A089022, A035610, A130976, A130977, A130978, A130979, A130980, A131521.

Rows n=0-3 give: A000012, A001477, A000384, A027849(k-1) for k>0.

Main diagonal gives A294491.

Coefficients of row polynomials in k, (k-1) are given by A157491, A039599.

Cf. A007318, A183134, A256116, A256117.

Programs

Maple

A:= proc(n, k) local j;
      if n=0 then 1
             else k/n *add(binomial(2*n,j) *(n-j) *(k-1)^j, j=0..n-1)
      fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

Mathematica

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 *)

Formula

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.

A(n,k) = A183134(n,k) if n=0 or k<2, A(n,k) = A183134(n,k)*k otherwise.

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.