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A242464 Number A(n,k) of n-length words w over a k-ary alphabet {a_1,...,a_k} such that w contains never more than j consecutive letters a_j (for 1<=j<=k); square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 3, 0, 0, 1, 4, 8, 4, 0, 0, 1, 5, 15, 21, 5, 0, 0, 1, 6, 24, 56, 54, 7, 0, 0, 1, 7, 35, 115, 208, 140, 9, 0, 0, 1, 8, 48, 204, 550, 773, 362, 12, 0, 0, 1, 9, 63, 329, 1188, 2631, 2872, 937, 16, 0, 0, 1, 10, 80, 496, 2254, 6919, 12584, 10672, 2425, 21, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
The sequence of column k satisfies a linear recurrence with constant coefficients of order A015614(k+1) for k>1.
LINKS
FORMULA
G.f. of column k: 1/(1-Sum_{i=1..k} v(i)/(1+v(i))) with v(i) = (x-x^(i+1))/(1-x).
EXAMPLE
A(0,k) = 1 for all k: the empty word.
A(1,5) = 5: [1], [2], [3], [4], [5].
A(2,4) = 15: [1,2], [1,3], [1,4], [2,1], [2,2], [2,3], [2,4], [3,1], [3,2], [3,3], [3,4], [4,1], [4,2], [4,3], [4,4].
A(3,3) = 21: [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [1,3,3], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2], [3,3,3].
A(4,2) = 5: [1,2,1,2], [1,2,2,1], [2,1,2,1], [2,1,2,2], [2,2,1,2].
A(n,1) = 0 for n>1.
A(n,0) = 0 for n>0.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 0, 3, 8, 15, 24, 35, 48, ...
0, 0, 4, 21, 56, 115, 204, 329, ...
0, 0, 5, 54, 208, 550, 1188, 2254, ...
0, 0, 7, 140, 773, 2631, 6919, 15443, ...
0, 0, 9, 362, 2872, 12584, 40295, 105804, ...
0, 0, 12, 937, 10672, 60191, 234672, 724892, ...
MAPLE
b:= proc(n, k, c, t) option remember;
`if`(n=0, 1, add(`if`(c=t and j=c, 0,
b(n-1, k, j, 1+`if`(j=c, t, 0))), j=1..k))
end:
A:= (n, k)-> b(n, k, 0$2):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
nn=10; Transpose[Map[PadRight[#, nn]&, Table[CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->(z-z^(i+1))/(1-z), {i, 1, n}]), {z, 0, nn}], z], {n, 0, nn}]]]//Grid
(* Second program: *)
b[n_, k_, c_, t_] := b[n, k, c, t] = If[n == 0, 1, Sum[If[c == t && j == c, 0, b[n - 1, k, j, 1 + If[j == c, t, 0]]], {j, 1, k}]];
A[n_, k_] := b[n, k, 0, 0];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2020, after Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A019590(n+1), A164001(n+1), A242452, A242495, A242509, A242629, A242630, A242631, A242632, A242633.
Rows n=0-2 give: A000012, A001477, A005563(k-1) for k>0.
Main diagonal gives A242635.
Sequence in context: A309021 A307968 A338501 * A273185 A351776 A259784
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved

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Last modified February 21 10:12 EST 2024. Contains 370228 sequences. (Running on oeis4.)