A183134 Square array A(n,k) by antidiagonals. A(n,k) is the number of length 2n k-ary words (n,k>=0), either empty or beginning with the first character of the alphabet, that can be built by repeatedly inserting doublets into the initially empty word.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 5, 10, 1, 0, 1, 1, 7, 29, 35, 1, 0, 1, 1, 9, 58, 181, 126, 1, 0, 1, 1, 11, 97, 523, 1181, 462, 1, 0, 1, 1, 13, 146, 1145, 4966, 7941, 1716, 1, 0, 1, 1, 15, 205, 2131, 14289, 48838, 54573, 6435, 1, 0

Offset

0,13

Comments

Column k > 2 is asymptotic to 2^(2*n) * (k-1)^(n+1) / ((k-2)^2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 07 2014

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra, arXiv preprint arXiv:1303.3481, 2013

A. Lakshminarayan, Z. Puchala, K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169, 2014

Formula

A(n,k) = 1 if n=0, A(n,k) = k if n>0 and k<=1, and A(n,k) = 1/n * Sum_{j=0..n-1} C(2*n,j)*(n-j)*(k-1)^j else.

A(n,k) = A183135(n,k) if n=0 or k<2, A(n,k) = A183135(n,k)/k else.

G.f. of column k: 1/(1-k*x) if k<2, (1-1/k) * (1 + 2 / (k-2 + k * sqrt (1-(4*k-4)*x))) else.

Example

A(3,2) = 10, because 10 words of length 6 beginning with the first character of the 2-letter alphabet {a, b} can be built by repeatedly inserting doublets (words with two equal letters) into the initially empty word: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, abaaba, abbaaa, abbabb, abbbba.
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,  ...
  0,  1,   1,    1,    1,     1,  ...
  0,  1,   3,    5,    7,     9,  ...
  0,  1,  10,   29,   58,    97,  ...
  0,  1,  35,  181,  523,  1145,  ...
  0,  1, 126, 1181, 4966, 14289,  ...

Maple

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

Mathematica

a[n_, k_] := If[ n == 0, 1 , If[ k <= 1, k, Sum [Binomial[2*n, j]*(n-j)*(k-1)^j, {j, 0, n-1}] / n ] ]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

Crossrefs

Rows 0-10 give: A000012, A057427, A004273, A079273(k) for k>0, A194716, A194717, A194718, A194719, A194720, A194721, A194722.

Columns 0-10 give: A000007, A000012, A001700(n-1) for n>0, A194723, A194724, A194725, A194726, A194727, A194728, A194729, A194730.

Main diagonal gives A248828.

Coefficients of row polynomials for k>0 in k, (k+1) are given by A050166, A157491.

Cf. A007318, A183135.

Sequence in context: A231345 A271344 A327622 * A328747 A346061 A053382

Adjacent sequences: A183131 A183132 A183133 * A183135 A183136 A183137

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 26 2010

Status

approved