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Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).
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%I #25 Feb 09 2024 01:44:17

%S 1,1,1,6,11,81,176,1406,3311,27636,68211,585162,1489488,13019909,

%T 33870540,300138696,793542167,7105216833,19022318084,171717015470,

%U 464333035881,4219267597578,11502251937176,105085831400550,288417894029200,2647012241261856,7306488667126803

%N Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

%C The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, this sequence, and A192894. From this one should be able to guess a g.f.

%C Number of achiral noncrossing partitions composed of n blocks of size 11. - _Andrew Howroyd_, Feb 08 2024

%H Andrew Howroyd, <a href="/A192893/b192893.txt">Table of n, a(n) for n = 0..500</a>

%H Michel Bousquet and Cédric Lamathe, <a href="https://doi.org/10.46298/dmtcs.420">On symmetric structures of order two</a>, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.

%F From _Andrew Howroyd_, Feb 08 2024: (Start)

%F a(2n) = binomial(11*n,n)/(10*n+1); a(2n+1) = binomial(11*n+5,n)*6/(10*n+6).

%F G.f. satisfies: A(x) = 1 + x*A(x)^6*A(-x)^5. (End)

%o (PARI) a(n)={my(m=n\2, p=5*(n%2)+1); binomial(11*m+p-1, m)*p/(10*m+p)} \\ _Andrew Howroyd_, Feb 08 2024

%Y Column k=11 of A369929 and k=12 of A370062.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Jul 12 2011

%E a(11) onwards from _Andrew Howroyd_, Jan 26 2024

%E a(0)=1 prepended by _Andrew Howroyd_, Feb 08 2024