OFFSET
0,4
COMMENTS
Number of achiral polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christersson link. - Robert A. Russell, Jan 23 2024
Number of achiral noncrossing partitions composed of n blocks of size 5. - Andrew Howroyd, Feb 08 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
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
Malin Christersson, Make hyperbolic tilings of images, web page, 2019.
FORMULA
G.f.: A(x) = G(x^2) + x*G(x^2)^3 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
a(2n) = binomial(5*n,n)/(4*n+1); a(2n+1) = binomial(5*n+2,n)*3/(4*n+3).
From Robert A. Russell, Jan 23 2024: (Start)
a(n+2)/a(n) ~ 3125/256. a(2m+1)/a(2m) ~ 75/16; a(2m)/a(2m-1) ~ 125/48.
a(2m) = A002294(m) ~ (5^5/4^4)^m*sqrt(5/(2*Pi*(4*m)^3)). - Robert A. Russell, Jul 15 2024
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)^5 is the g.f. of A002294.
a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+2*j+2*k=n-1} a(i) * a(2*j) * a(2*k). (End)
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (-1)^(i+j) * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025
a(n) ~ c * 5^(5*n/2) / (4^(2*n+1) * n^(3/2) * sqrt(Pi)), where c = 3 if n is odd and c = sqrt(5) if n is even. - Amiram Eldar, Sep 16 2025
a(2m+1) = A118970(m). - Robert A. Russell, Oct 16 2025
D-finite with recurrence: 234375*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 15000*(11250*n^4 + 78750*n^3 + 194500*n^2 + 205025*n + 78777)*a(n + 1) + 320*(10000*n^4 + 382500*n^3 + 2287625*n^2 + 4948125*n + 3656214)*a(n + 2) - 76800*(2*n + 7)*(n + 3)*(90*n^2 + 315*n + 244)*a(n + 3) - 311296*(n + 4)*(n + 3)*(2*n + 9)*(2*n + 7)*a(n + 4) = 0. - Robert Israel, Mar 19 2026
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 18*x^5 + 35*x^6 + 136*x^7 + ...
A(x) = 1 + x*A(x)^3*A(-x)^2 where
A(x)^3 = 1 + 3x + 6x^2 + 16x^3 + 39x^4 + 114x^5 + 304x^6 + 936x^7 + ...
A(-x)^2 = 1 - 2x + 3x^2 - 8x^3 + 17x^4 - 52x^5 + 125x^6 - 408x^7 + ...
Also, A(x) = G(x^2) + x*G(x^2)^3 where
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + ...
G(x)^3 = 1 + 3*x + 18*x^2 + 136*x^3 + 1155*x^4 + 10530*x^5 + ...
MAPLE
f:= rectoproc({234375*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*a(n) + 15000*(11250*n^4 + 78750*n^3 + 194500*n^2 + 205025*n + 78777)*a(n + 1) + 320*(10000*n^4 + 382500*n^3 + 2287625*n^2 + 4948125*n + 3656214)*a(n + 2) - 76800*(2*n + 7)*(n + 3)*(90*n^2 + 315*n + 244)*a(n + 3) - 311296*(n + 4)*(n + 3)*(2*n + 9)*(2*n + 7)*a(n + 4), a(0)=1, a(1)=1, a(2)=1, a(3)=3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 19 2026
MATHEMATICA
terms = 28;
A[_] = 1; Do[A[x_] = 1 + x A[x]^3 A[-x]^2 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)
p=6; Table[If[EvenQ[n], Binomial[(p-1)n/2, n/2]/((p-2)n/2+1), If[OddQ[p], (p-1)Binomial[(p-1)n/2-1, (n-1)/2]/((p-2)n+1), p Binomial[(p-1)n/2-1/2, (n-1)/2]/((p-2)n+2)]], {n, 0, 35}] (* Robert A. Russell, Jan 23 2024 *)
PROG
(PARI) {a(n)=my(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^3*subst(A^2, x, -x)); polcoef(A, n)}
(PARI) {a(n)=my(m=n\2, p=2*(n%2)+1); binomial(5*m+p-1, m)*p/(4*m+p)}
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Aug 23 2008
STATUS
approved