OFFSET
0,4
COMMENTS
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.
Number of achiral noncrossing partitions composed of n blocks of size 11. - Andrew Howroyd, Feb 08 2024
Number of achiral polyominoes composed of n dodecagonal cells of the hyperbolic regular tiling with Schläfli symbol {12,oo}. - Robert A. Russell, Oct 02 2025
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.
FORMULA
From Andrew Howroyd, Feb 08 2024: (Start)
a(2n) = binomial(11*n,n)/(10*n+1); a(2n+1) = binomial(11*n+5,n)*6/(10*n+6).
G.f. A(x) satisfies A(x) = 1 + x*A(x)^6*A(-x)^5. (End)
From Seiichi Manyama, Jul 07 2025: (Start)
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.
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)
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
a(n) ~ c * 11^(11*n/2) / (2^(5*n+1/2) * 5^(5*n+3/2) * n^(3/2) * sqrt(Pi)), where c = 6 if n is odd and c = sqrt(11) if n is even. - Amiram Eldar, Sep 16 2025
MATHEMATICA
a[n_] := If[EvenQ[n], Binomial[11*n/2, n/2]/(5*n + 1), Binomial[(11*n - 1)/2, (n - 1)/2]*6/(5*n + 1)]; Array[a, 27, 0] (* Amiram Eldar, Sep 16 2025 *)
Table[If[OddQ[n], 6Binomial[(11n-1)/2, 5n], Binomial[11n/2, 5n]]/(5n+1), {n, 0, 40}] (* Robert A. Russell, Oct 02 2025 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 12 2011
EXTENSIONS
a(11) onwards from Andrew Howroyd, Jan 26 2024
a(0)=1 prepended by Andrew Howroyd, Feb 08 2024
STATUS
approved