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A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

%I A192894 #36 Jul 09 2025 10:14:56
%S A192894 1,1,1,7,13,112,247,2310,5525,53998,135408,1360289,3518515,36017352,
%T A192894 95223414,988172368,2655417765,27844071255,75769712590,801012669457,
%U A192894 2201663313200,23428926096576,64924369564353,694644371065372,1938034271677595,20829931845958872,58448142042957576
%N A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).
%C A192894 The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, A192893, A192894. From this one should be able to guess a g.f.
%H A192894 Andrew Howroyd, <a href="/A192894/b192894.txt">Table of n, a(n) for n = 0..200</a>
%H A192894 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 A192894 From _Seiichi Manyama_, Jul 07 2025: (Start)
%F A192894 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).
%F A192894 G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^13.
%F A192894 a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+2*(x_2+x_3+...+x_7)=n-1} a(x_1) * Product_{k=2..7} a(2*x_k). (End)
%F A192894 a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_13>=0 and x_1+x_2+...+x_13=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5+x_6) * Product_{k=1..13} a(x_k). - _Seiichi Manyama_, Jul 09 2025
%Y A192894 Column k=13 of A369929 and k=14 of A370062.
%Y A192894 Cf. A143049.
%K A192894 nonn
%O A192894 0,4
%A A192894 _N. J. A. Sloane_, Jul 12 2011
%E A192894 a(11) onwards from _Andrew Howroyd_, Jan 26 2024
%E A192894 a(0)=1 prepended by _Seiichi Manyama_, Jul 07 2025