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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 1..140, flattened

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-Fran├žois Alcover, Feb 22 2017, translated from Maple *)

CROSSREFS

Rows n=0-2 give: A000012, A001477, A000567.

Columns k=0-2 give: A000007, A000012, A047098.

Cf. A183134, A183135, A213027, A256311.

Sequence in context: A322836 A305466 A160114 * A287698 A109970 A116088

Adjacent sequences:  A213025 A213026 A213027 * A213029 A213030 A213031

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 03 2012

STATUS

approved

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Last modified September 19 11:06 EDT 2019. Contains 327192 sequences. (Running on oeis4.)