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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.

%I A340156 #73 Feb 03 2021 23:01:36
%S A340156 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,
%T A340156 991,963,201,1,15,133,751,2569,4612,3217,423,1,17,176,1219,5531,15085,
%U A340156 20905,10547,880,1,19,225,1849,10513,39186,86241,92935,34089,1815
%N A340156 Square array read by upward antidiagonals: T(n, k) is the number of n-ary strings of length k containing 00.
%H A340156 Robert P. P. McKone, <a href="/A340156/b340156.txt">Antidiagonals n = 2..100, flattened</a>
%F A340156 T(n, k) = n^k - A180165(n+1,k-1), where A180165 in the number of strings not containing 00.
%F A340156 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".
%e A340156 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}.
%e A340156 Square table T(n,k):
%e A340156      k=2:  k=3:  k=4:   k=5:    k=6:     k=7:
%e A340156 n=2:   1     3     8     19      43       94
%e A340156 n=3:   1     5    21     79     281      963
%e A340156 n=4:   1     7    40    205     991     4612
%e A340156 n=5:   1     9    65    421    2569    15085
%e A340156 n=6:   1    11    96    751    5531    39186
%e A340156 n=7:   1    13   133   1219   10513    87199
%e A340156 n=8:   1    15   176   1849   18271   173608
%e A340156 n=9:   1    17   225   2665   29681   317817
%t A340156 m[r_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]];
%t A340156 T[n_, k_, r_] := MatrixPower[m[r], k][[1, r + 1]]*n^k;
%t A340156 Reverse[Table[T[n, k - n + 2, 2], {k, 2, 11}, {n, 2, k}], 2] // Flatten (* _Robert P. P. McKone_, Jan 26 2021 *)
%Y A340156 Cf. A008466 (row 2), A186244 (row 3), A000567 (column 4).
%Y A340156 Cf. A180165 (not containing 00), A340242 (containing 000).
%Y A340156 Cf. A000045, A028859, A125145, A086347, A180033, A180167, A322054.
%Y A340156 Cf. A005563, A033445.
%K A340156 nonn,tabl
%O A340156 2,3
%A A340156 _Robert P. P. McKone_, Dec 29 2020