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A342239 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that are bifix free; n >= 1, k >= 1.

%I A342239 #22 Apr 18 2022 12:31:22
%S A342239 1,2,0,3,2,0,4,6,4,0,5,12,18,6,0,6,20,48,48,12,0,7,30,100,180,144,20,
%T A342239 0,8,42,180,480,720,414,40,0,9,56,294,1050,2400,2832,1242,74,0,10,72,
%U A342239 448,2016,6300,11900,11328,3678,148,0,11,90,648,3528,14112,37620,59500,45132,11034,284,0
%N A342239 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that are bifix free; n >= 1, k >= 1.
%H A342239 Peter Kagey, <a href="/A342239/b342239.txt">Antidiagonals n = 1..100, flattened</a>
%F A342239 T(n,0)    = n.
%F A342239 T(n,2k)   = n*T(n,2k-1) - T(n,k).
%F A342239 T(n,2k+1) = n*T(n,2k).
%e A342239 Table begins:
%e A342239 n\k | 1  2   3    4     5      6       7        8         9
%e A342239 ----+------------------------------------------------------
%e A342239   1 | 1  0   0    0     0      0       0        0         0
%e A342239   2 | 2  2   4    6    12     20      40       74       148
%e A342239   3 | 3  6  18   48   144    414    1242     3678     11034
%e A342239   4 | 4 12  48  180   720   2832   11328    45132    180528
%e A342239   5 | 5 20 100  480  2400  11900   59500   297020   1485100
%e A342239   6 | 6 30 180 1050  6300  37620  225720  1353270   8119620
%e A342239   7 | 7 42 294 2016 14112  98490  689430  4823994  33767958
%e A342239   8 | 8 56 448 3528 28224 225344 1802752 14418488 115347904
%Y A342239 Rows: A003000 (n=2), A019308 (n=3), A019309 (n=4).
%Y A342239 Columns: A002378 (k=1), A045991 (k=2), A047927 (k=3).
%K A342239 nonn,tabl
%O A342239 1,2
%A A342239 _Peter Kagey_, Mar 06 2021