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A143546 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x)^2.

%I A143546 #60 Jul 08 2025 07:41:47
%S A143546 1,1,1,3,5,18,35,136,285,1155,2530,10530,23751,100688,231880,996336,
%T A143546 2330445,10116873,23950355,104819165,250543370,1103722620,2658968130,
%U A143546 11777187240,28558343775,127067830773,309831575760,1383914371728,3390416787880,15194457001440
%N A143546 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x)^2.
%C A143546 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 Christensson link. - _Robert A. Russell_, Jan 23 2024
%C A143546 Number of achiral noncrossing partitions composed of n blocks of size 5. - _Andrew Howroyd_, Feb 08 2024
%H A143546 Andrew Howroyd, <a href="/A143546/b143546.txt">Table of n, a(n) for n = 0..1000</a>
%H A143546 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. - From _N. J. A. Sloane_, Jul 12 2011
%H A143546 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%F A143546 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.
%F A143546 a(2n) = binomial(5*n,n)/(4*n+1); a(2n+1) = binomial(5*n+2,n)*3/(4*n+3).
%F A143546 From _Robert A. Russell_, Jan 23 2024: (Start)
%F A143546 a(n+2)/a(n) ~ 3125/256. a(2m+1)/a(2m) ~ 75/16; a(2m)/a(2m-1) ~ 125/48.
%F A143546 a(n) = 2*A004127(n) - A221184(n-1) = A221184(n-1) - 2*A369473(n) = A004127(n) - A369473(n). (End)
%F A143546 a(2m) = A002294(m) ~ (5^5/4^4)^m*sqrt(5/(2*Pi*(4*m)^3)). - _Robert A. Russell_, Jul 15 2024
%F A143546 From _Seiichi Manyama_, Jul 07 2025: (Start)
%F A143546 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.
%F A143546 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)
%F A143546 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
%e A143546 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 18*x^5 + 35*x^6 + 136*x^7 + ...
%e A143546 A(x) = 1 + x*A(x)^3*A(-x)^2 where
%e A143546 A(x)^3 = 1 + 3x + 6x^2 + 16x^3 + 39x^4 + 114x^5 + 304x^6 + 936x^7 + ...
%e A143546 A(-x)^2 = 1 - 2x + 3x^2 - 8x^3 + 17x^4 - 52x^5 + 125x^6 - 408x^7 + ...
%e A143546 Also, A(x) = G(x^2) + x*G(x^2)^3 where
%e A143546 G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + ...
%e A143546 G(x)^3 = 1 + 3*x + 18*x^2 + 136*x^3 + 1155*x^4 + 10530*x^5 + ...
%t A143546 terms = 28;
%t A143546 A[_] = 1; Do[A[x_] = 1 + x A[x]^3 A[-x]^2 + O[x]^terms // Normal, {terms}];
%t A143546 CoefficientList[A[x], x] (* _Jean-François Alcover_, Jul 24 2018 *)
%t A143546 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 *)
%o A143546 (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)}
%o A143546 (PARI) {a(n)=my(m=n\2,p=2*(n%2)+1);binomial(5*m+p-1,m)*p/(4*m+p)}
%Y A143546 Column k=5 of A369929 and k=6 of A370062.
%Y A143546 Cf. A118970.
%Y A143546 Polyominoes: A221184(n-1) (oriented), A004127 (unoriented), A369473 (chiral), A002294 (rooted), A047749 {4,oo}, A369472 {5,oo}.
%K A143546 nonn,easy
%O A143546 0,4
%A A143546 _Paul D. Hanna_, Aug 23 2008