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A340156 Square array read by upward antidiagonals: T(n, k) is the number of n-ary strings of length k containing 00. 5
1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 9, 40, 79, 43, 1, 11, 65, 205, 281, 94, 1, 13, 96, 421, 991, 963, 201, 1, 15, 133, 751, 2569, 4612, 3217, 423, 1, 17, 176, 1219, 5531, 15085, 20905, 10547, 880, 1, 19, 225, 1849, 10513, 39186, 86241, 92935, 34089, 1815 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,3
LINKS
FORMULA
T(n, k) = n^k - A180165(n+1,k-1), where A180165 in the number of strings not containing 00.
m(2) = [1 - 1/n, 1/n, 0; 1 - 1/n, 0, 1/n; 0, 0, 1], is the probability/transition matrix for two consecutive "0" -> "containing 00".
EXAMPLE
For n = 3 and k = 4, there are 21 strings: {0000, 0001, 0002, 0010, 0011, 0012, 0020, 0021, 0022, 0100, 0200, 1000, 1001, 1002, 1100, 1200, 2000, 2001, 2002, 2100, 2200}.
Square table T(n,k):
k=2: k=3: k=4: k=5: k=6: k=7:
n=2: 1 3 8 19 43 94
n=3: 1 5 21 79 281 963
n=4: 1 7 40 205 991 4612
n=5: 1 9 65 421 2569 15085
n=6: 1 11 96 751 5531 39186
n=7: 1 13 133 1219 10513 87199
n=8: 1 15 176 1849 18271 173608
n=9: 1 17 225 2665 29681 317817
MATHEMATICA
m[r_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]];
T[n_, k_, r_] := MatrixPower[m[r], k][[1, r + 1]]*n^k;
Reverse[Table[T[n, k - n + 2, 2], {k, 2, 11}, {n, 2, k}], 2] // Flatten (* Robert P. P. McKone, Jan 26 2021 *)
CROSSREFS
Cf. A008466 (row 2), A186244 (row 3), A000567 (column 4).
Cf. A180165 (not containing 00), A340242 (containing 000).
Sequence in context: A038738 A210741 A208760 * A340242 A116647 A063858
KEYWORD
nonn,tabl
AUTHOR
Robert P. P. McKone, Dec 29 2020
STATUS
approved

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Last modified March 28 11:46 EDT 2024. Contains 371241 sequences. (Running on oeis4.)