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A341050 Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.

%I A341050 #38 Aug 05 2023 15:54:38
%S A341050 1,1,1,3,1,1,3,1,5,8,1,1,3,1,5,8,1,7,21,19,1,1,3,1,5,8,1,7,21,20,1,9,
%T A341050 40,81,43,1,1,3,1,5,8,1,7,21,20,1,9,40,81,47,1,11,65,208,295,94,1,1,3,
%U A341050 1,5,8,1,7,21,20,1,9,40,81,48,1,11,65,208,297,107,1,13,96,425,1024,1037,201
%N A341050 Cube array read by upward antidiagonals ignoring zero and empty terms: T(n, k, r) is the number of n-ary strings of length k, containing r consecutive 0's.
%H A341050 Robert P. P. McKone, <a href="/A341050/b341050.txt">Antidiagonals n = 2..50, flattened</a>
%e A341050 For n = 5, k = 6 and r = 4, there are 65 strings: {000000, 000001, 000002, 000003, 000004, 000010, 000011, 000012, 000013, 000014, 000020, 000021, 000022, 000023, 000024, 000030, 000031, 000032, 000033, 000034, 000040, 000041, 000042, 000043, 000044, 010000, 020000, 030000, 040000, 100000, 100001, 100002, 100003, 100004, 110000, 120000, 130000, 140000, 200000, 200001, 200002, 200003, 200004, 210000, 220000, 230000, 240000, 300000, 300001, 300002, 300003, 300004, 310000, 320000, 330000, 340000, 400000, 400001, 400002, 400003, 400004, 410000, 420000, 430000, 440000}
%e A341050 The first seven slices of the tetrahedron (or pyramid) are:
%e A341050 -----------------Slice 1-----------------
%e A341050   1
%e A341050 -----------------Slice 2-----------------
%e A341050     1
%e A341050   1  3
%e A341050 -----------------Slice 3-----------------
%e A341050       1
%e A341050     1  3
%e A341050   1  5  8
%e A341050 -----------------Slice 4-----------------
%e A341050         1
%e A341050       1  3
%e A341050     1  5   8
%e A341050   1  7  21  19
%e A341050 -----------------Slice 5-----------------
%e A341050           1
%e A341050         1  3
%e A341050       1  5   8
%e A341050     1  7  21  20
%e A341050   1  9  40  81  43
%e A341050 -----------------Slice 6-----------------
%e A341050               1
%e A341050            1    3
%e A341050         1    5     8
%e A341050       1   7    21    20
%e A341050     1   9   40    81    47
%e A341050   1  11  65   208   295   94
%e A341050 -----------------Slice 7-----------------
%e A341050                  1
%e A341050               1     3
%e A341050            1     5     8
%e A341050          1    7     21    20
%e A341050       1    9    40     81      48
%e A341050     1   11   65    208     297     107
%e A341050   1  13   96   425    1024    1037    201
%t A341050 m[r_, n_] := 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, n], k][[1, r + 1]]*n^k; DeleteCases[Transpose[PadLeft[Reverse[Table[T[n, k, r], {k, 2, 8}, {r, 2, k}, {n, 2, r}], 2]], 2 <-> 3], 0, 3] // Flatten
%Y A341050 Cf. A340156 (r=2), A340242 (r=3).
%Y A341050 Cf. A008466 (n=2, r=2), A186244 (n=3, r=2), A050231 (n=2, r=3), A231430 (n=3, r=3).
%Y A341050 Cf. A005408, A003215, A005917, A022521, A022522, A022523, A022524, A022525, A022526, A022527, A022528, A022529, A022530, A022531, A022532, A022533, A022534, A022535, A022536, A022537, A022538, A022539, A022540 (k=x, r=1, where x is the x-th Nexus Number).
%Y A341050 Cf. A000567 [(k=4, r=2),(k=5, r=3),(k=6, r=4),...,(k=x, r=x-2)].
%Y A341050 Cf. A103532 [(k=6, r=3),(k=7, r=4),(k=8, r=5),...,(k=x, r=x-3)].
%K A341050 nonn,look,tabf
%O A341050 2,4
%A A341050 _Robert P. P. McKone_, Feb 04 2021