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A327873 Irregular triangle read by rows: T(n,k) is the number of length n primitive (aperiodic) palindromes using exactly k different symbols, 1 <= k <= ceiling(n/2). 8

%I

%S 1,0,0,2,0,2,0,6,6,0,4,6,0,14,36,24,0,12,36,24,0,28,150,240,120,0,24,

%T 144,240,120,0,62,540,1560,1800,720,0,54,534,1560,1800,720,0,126,1806,

%U 8400,16800,15120,5040,0,112,1770,8376,16800,15120,5040

%N Irregular triangle read by rows: T(n,k) is the number of length n primitive (aperiodic) palindromes using exactly k different symbols, 1 <= k <= ceiling(n/2).

%H Andrew Howroyd, <a href="/A327873/b327873.txt">Table of n, a(n) for n = 1..2550</a>

%F T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A284823(n,j).

%F T(n,k) = Sum_{d|n} mu(n/d)*k!*Stirling2(ceiling(d/2), k).

%e Triangle begins:

%e 1;

%e 0;

%e 0, 2;

%e 0, 2;

%e 0, 6, 6;

%e 0, 4, 6;

%e 0, 14, 36, 24;

%e 0, 12, 36, 24;

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

%e 0, 24, 144, 240, 120;

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

%e 0, 54, 534, 1560, 1800, 720;

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

%e 0, 112, 1770, 8376, 16800, 15120, 5040;

%e ...

%o (PARI) T(n,k) = {sumdiv(n, d, moebius(n/d)*k!*stirling(ceil(d/2), k, 2))}

%Y Columns k=2..6 are A056463, A056464, A056465, A056466, A056467.

%Y Row sums are A327874.

%Y Cf. A284823, A327878.

%K nonn,tabf

%O 1,4

%A _Andrew Howroyd_, Sep 28 2019

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Last modified January 26 20:42 EST 2020. Contains 331288 sequences. (Running on oeis4.)