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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). 8

%I

%S 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 240,120,0,39,339,900,960,360,0,62,540,1560,1800,720,0,81,1149,4968,

%U 9300,7920,2520,0,126,1806,8400,16800,15120,5040,0,175,3765,24588,71400,103320,73080,20160

%N 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 Primitive periodic palindromes may also be called achiral Lyndon words.

%H Andrew Howroyd, <a href="/A327878/b327878.txt">Table of n, a(n) for n = 1..2600</a>

%F T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A284856(n,j).

%F Column k is the Moebius transform of column k of A305540.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 2;

%e 0, 3, 3;

%e 0, 6, 6;

%e 0, 7, 21, 12;

%e 0, 14, 36, 24;

%e 0, 18, 90, 132, 60;

%e 0, 28, 150, 240, 120;

%e 0, 39, 339, 900, 960, 360;

%e 0, 62, 540, 1560, 1800, 720;

%e 0, 81, 1149, 4968, 9300, 7920, 2520;

%e 0, 126, 1806, 8400, 16800, 15120, 5040;

%e 0, 175, 3765, 24588, 71400, 103320, 73080, 20160;

%e ...

%o (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 Columns k=2..6 are A056498, A056499, A056500, A056501, A056502.

%Y Row sums are A327879.

%Y Cf. A284856, A305540, A327873.

%K nonn,tabf

%O 1,5

%A _Andrew Howroyd_, Sep 28 2019

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Last modified April 3 22:48 EDT 2020. Contains 333207 sequences. (Running on oeis4.)