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A342237 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that begin with a palindrome of two or more letters; n, k >= 1.

%I A342237 #24 Apr 18 2022 12:30:56
%S A342237 0,0,1,0,2,1,0,3,6,1,0,4,15,14,1,0,5,28,51,30,1,0,6,45,124,165,62,1,0,
%T A342237 7,66,245,532,507,126,1,0,8,91,426,1305,2164,1551,254,1,0,9,120,679,
%U A342237 2706,6605,8788,4683,510,1,0,10,153,1016,5005,16386,33405,35284,14127,1022,1
%N A342237 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that begin with a palindrome of two or more letters; n, k >= 1.
%H A342237 Peter Kagey, <a href="/A342237/b342237.txt">Antidiagonals n = 1..100, flattened</a>
%F A342237 T(n,1)    = 0.
%F A342237 T(n,2k)   = n*T(n,2k-1) + n^k - T(n,k).
%F A342237 T(n,2k+1) = n*T(n,2k) + n^(k+1) - T(n,k+1).
%e A342237 Table begins:
%e A342237 n\k | 1  2   3    4      5       6        7         8
%e A342237 ----+------------------------------------------------
%e A342237   1 | 0  1   1    1      1       1        1         1
%e A342237   2 | 0  2   6   14     30      62      126       254
%e A342237   3 | 0  3  15   51    165     507     1551      4683
%e A342237   4 | 0  4  28  124    532    2164     8788     35284
%e A342237   5 | 0  5  45  245   1305    6605    33405    167405
%e A342237   6 | 0  6  66  426   2706   16386    99186    595986
%e A342237   7 | 0  7  91  679   5005   35287   248731   1742839
%e A342237   8 | 0  8 120 1016   8520   68552   551496   4415048
%Y A342237 Rows: A000918 (n=2), A248122 (n=3), A249629 (n=4), A249638 (n=5), A249639 (n=6), A249640 (n=7), A249641 (n=8), A249642 (n=9), A249643 (n=10).
%Y A342237 Columns: A000384 (k=3), A007588 (k=4).
%K A342237 nonn,tabl
%O A342237 1,5
%A A342237 _Peter Kagey_, Mar 06 2021