A213028 - OEIS

%I #24 Jan 17 2024 17:40:27

%S 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,

%T 65,508,1773,1062,1,0,1,7,96,1085,7240,18303,5948,1,0,1,8,133,1986,

%U 20425,110524,197157,34120,1,0,1,9,176,3283,46476,412965,1766416,2189799,199316,1,0

%N 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.

%H Alois P. Heinz, <a href="/A213028/b213028.txt">Antidiagonals n = 1..140, flattened</a>

%F 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.

%F A(n,k) = k * A213027(n,k) if n>0, k>1; else A(n,k) = A213027(n,k).

%e A(0,k) = 1: the empty word.

%e A(n,1) = 1: (aaa)^n.

%e 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.

%e A(1,3) = 3: aaa, bbb, ccc.

%e 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.

%e Square array A(n,k) begins:

%e   1, 1,    1,      1,       1,       1,        1, ...

%e   0, 1,    2,      3,       4,       5,        6, ...

%e   0, 1,    8,     21,      40,      65,       96, ...

%e   0, 1,   38,    183,     508,    1085,     1986, ...

%e   0, 1,  196,   1773,    7240,   20425,    46476, ...

%e   0, 1, 1062,  18303,  110524,  412965,  1170066, ...

%e   0, 1, 5948, 197157, 1766416, 8755985, 30921756, ...

%p A:= (n, k)-> `if`(n=0, 1,

%p     k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%t 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 *)

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

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

%Y Cf. A183134, A183135, A213027, A256311.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Jun 03 2012