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A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

%I A192893 #39 Jul 09 2025 10:15:00
%S A192893 1,1,1,6,11,81,176,1406,3311,27636,68211,585162,1489488,13019909,
%T A192893 33870540,300138696,793542167,7105216833,19022318084,171717015470,
%U A192893 464333035881,4219267597578,11502251937176,105085831400550,288417894029200,2647012241261856,7306488667126803
%N A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).
%C A192893 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 A192893 Number of achiral noncrossing partitions composed of n blocks of size 11. - _Andrew Howroyd_, Feb 08 2024
%H A192893 Andrew Howroyd, <a href="/A192893/b192893.txt">Table of n, a(n) for n = 0..500</a>
%H A192893 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 A192893 From _Andrew Howroyd_, Feb 08 2024: (Start)
%F A192893 a(2n) = binomial(11*n,n)/(10*n+1); a(2n+1) = binomial(11*n+5,n)*6/(10*n+6).
%F A192893 G.f. A(x) satisfies A(x) = 1 + x*A(x)^6*A(-x)^5. (End)
%F A192893 From _Seiichi Manyama_, Jul 07 2025: (Start)
%F A192893 G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^11 is the g.f. of A230388.
%F A192893 a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+2*(x_2+x_3+...+x_6)=n-1} a(x_1) * Product_{k=2..6} a(2*x_k). (End)
%F A192893 a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_11>=0 and x_1+x_2+...+x_11=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5) * Product_{k=1..11} a(x_k). - _Seiichi Manyama_, Jul 09 2025
%o A192893 (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 A192893 Column k=11 of A369929 and k=12 of A370062.
%Y A192893 Cf. A143048.
%K A192893 nonn
%O A192893 0,4
%A A192893 _N. J. A. Sloane_, Jul 12 2011
%E A192893 a(11) onwards from _Andrew Howroyd_, Jan 26 2024
%E A192893 a(0)=1 prepended by _Andrew Howroyd_, Feb 08 2024