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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.
4
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, 65, 508, 1773, 1062, 1, 0, 1, 7, 96, 1085, 7240, 18303, 5948, 1, 0, 1, 8, 133, 1986, 20425, 110524, 197157, 34120, 1, 0, 1, 9, 176, 3283, 46476, 412965, 1766416, 2189799, 199316, 1, 0
OFFSET
0,8
LINKS
FORMULA
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.
A(n,k) = k * A213027(n,k) if n>0, k>1; else A(n,k) = A213027(n,k).
EXAMPLE
A(0,k) = 1: the empty word.
A(n,1) = 1: (aaa)^n.
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.
A(1,3) = 3: aaa, bbb, ccc.
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.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 8, 21, 40, 65, 96, ...
0, 1, 38, 183, 508, 1085, 1986, ...
0, 1, 196, 1773, 7240, 20425, 46476, ...
0, 1, 1062, 18303, 110524, 412965, 1170066, ...
0, 1, 5948, 197157, 1766416, 8755985, 30921756, ...
MAPLE
A:= (n, k)-> `if`(n=0, 1,
k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
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 *)
CROSSREFS
Rows n=0-2 give: A000012, A001477, A000567.
Columns k=0-2 give: A000007, A000012, A047098.
Sequence in context: A322836 A305466 A160114 * A287698 A366834 A109970
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 03 2012
STATUS
approved