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A143546
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G.f. satisfies: A(x) = 1 + x*A(x)^3*A(-x)^2.
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17
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1, 1, 1, 3, 5, 18, 35, 136, 285, 1155, 2530, 10530, 23751, 100688, 231880, 996336, 2330445, 10116873, 23950355, 104819165, 250543370, 1103722620, 2658968130, 11777187240, 28558343775, 127067830773, 309831575760, 1383914371728, 3390416787880, 15194457001440
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OFFSET
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0,4
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COMMENTS
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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
Number of achiral noncrossing partitions composed of n blocks of size 5. - Andrew Howroyd, Feb 08 2024
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LINKS
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FORMULA
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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).
a(n+2)/a(n) ~ 3125/256. a(2m+1)/a(2m) ~ 75/16; a(2m)/a(2m-1) ~ 125/48.
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EXAMPLE
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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 + ...
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MATHEMATICA
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terms = 28;
A[_] = 1; Do[A[x_] = 1 + x A[x]^3 A[-x]^2 + O[x]^terms // Normal, {terms}];
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 *)
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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