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A327878 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromes using exactly k different symbols, 1 <= k <= 1 + floor(n/2).

%I A327878 #10 Oct 03 2019 20:59:38
%S A327878 1,0,1,0,2,0,3,3,0,6,6,0,7,21,12,0,14,36,24,0,18,90,132,60,0,28,150,
%T A327878 240,120,0,39,339,900,960,360,0,62,540,1560,1800,720,0,81,1149,4968,
%U A327878 9300,7920,2520,0,126,1806,8400,16800,15120,5040,0,175,3765,24588,71400,103320,73080,20160
%N A327878 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromes using exactly k different symbols, 1 <= k <= 1 + floor(n/2).
%C A327878 Primitive periodic palindromes may also be called achiral Lyndon words.
%H A327878 Andrew Howroyd, <a href="/A327878/b327878.txt">Table of n, a(n) for n = 1..2600</a>
%F A327878 T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A284856(n,j).
%F A327878 Column k is the Moebius transform of column k of A305540.
%e A327878 Triangle begins:
%e A327878   1;
%e A327878   0,   1;
%e A327878   0,   2;
%e A327878   0,   3,    3;
%e A327878   0,   6,    6;
%e A327878   0,   7,   21,    12;
%e A327878   0,  14,   36,    24;
%e A327878   0,  18,   90,   132,    60;
%e A327878   0,  28,  150,   240,   120;
%e A327878   0,  39,  339,   900,   960,    360;
%e A327878   0,  62,  540,  1560,  1800,    720;
%e A327878   0,  81, 1149,  4968,  9300,   7920,  2520;
%e A327878   0, 126, 1806,  8400, 16800,  15120,  5040;
%e A327878   0, 175, 3765, 24588, 71400, 103320, 73080, 20160;
%e A327878   ...
%o A327878 (PARI) T(n,k) = {sumdiv(n, d, moebius(n/d) * k! * (stirling((d+1)\2,k,2) + stirling(d\2+1,k,2)))/2}
%Y A327878 Columns k=2..6 are A056498, A056499, A056500, A056501, A056502.
%Y A327878 Row sums are A327879.
%Y A327878 Cf. A284856, A305540, A327873.
%K A327878 nonn,tabf
%O A327878 1,5
%A A327878 _Andrew Howroyd_, Sep 28 2019