OFFSET
0,4
COMMENTS
Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024
Number of achiral polyominoes composed of n decagonal cells of the hyperbolic regular tiling with Schläfli symbol {10,oo}. An achiral polyomino has an axis of symmetry. - Robert A. Russell, Oct 10 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. - From N. J. A. Sloane, Jul 12 2011
FORMULA
G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A059967.
a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+2*j+2*k+2*l+2*m=n-1} a(i) * a(2*j) * a(2*k) * a(2*l) * a(2*m). - Seiichi Manyama, Jul 07 2025
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (-1)^(x_1+x_2+x_3+x_4) * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 09 2025
a(n) ~ c * 3^(9*n/2) / (2^(12*n+7/2) * n^(3/2) * sqrt(Pi)), where c = 5 if n is odd and c = 3 if n is even. - Amiram Eldar, Sep 16 2025
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A059967, then
G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...
MATHEMATICA
terms = 25;
A[_] = 1; Do[A[x_] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)
Table[If[OddQ[n], 5Binomial[(9n-1)/2, 4n], Binomial[9n/2, 4n]]/(4n+1), {n, 0, 40}] (* Robert A. Russell, Oct 10 2025 *)
PROG
(PARI) {a(n)=my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^5*subst(A^4, x, -x)); polcoef(A, n)}
(PARI) {a(n)=my(m=n\2, p=4*(n%2)+1); binomial(9*m+p-1, m)*p/(8*m+p)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 24 2008
STATUS
approved