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

1, 1, 1, 7, 13, 112, 247, 2310, 5525, 53998, 135408, 1360289, 3518515, 36017352, 95223414, 988172368, 2655417765, 27844071255, 75769712590, 801012669457, 2201663313200, 23428926096576, 64924369564353, 694644371065372, 1938034271677595, 20829931845958872, 58448142042957576

Offset: 0

Author

N. J. A. Sloane, Jul 12 2011

Keywords

• nonn

Comments

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.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.

Crossrefs

Column k=13 of A369929 and k=14 of A370062.

Cf. A143049.

Formula

From Seiichi Manyama, Jul 07 2025: (Start)

G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).

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.

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)

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

Extensions

a(11) onwards from Andrew Howroyd, Jan 26 2024

a(0)=1 prepended by Seiichi Manyama, Jul 07 2025