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