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A089073
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Number of symmetric non-crossing connected graphs on n equidistant nodes on a circle.
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0
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1, 1, 2, 5, 10, 32, 64, 231, 462, 1792, 3584, 14586, 29172, 122880, 245760, 1062347, 2124694, 9371648, 18743296, 84021990, 168043980, 763363328, 1526726656, 7012604550, 14025209100, 65028489216, 130056978432, 607892634420
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OFFSET
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1,3
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COMMENTS
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Number of symmetric non-crossing connected graphs on n equidistant nodes on a circle (it is assumed that the axis of symmetry is a diameter of the circle passing through a given node). Example: a(4)=5 because on the nodes A,B,C,D (axis of symmetry through A) the only symmetric non-crossing connected graphs are {AB,AC,AD), (AC,BC,DC), (AB,BC,CD,DA), (AB,BC,CD,DA,BD), (AB,BC,CD,DA,AC).
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LINKS
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FORMULA
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a(2k) = 4^k*binomial((3k-1)/2, k)/[2(k+1)], a(2k+1) = 2a(2k).
Conjecture D-finite with recurrence n*(n+2)*(23*n^2-162*n+199) *a(n) +12*(27*n^2-47*n-10) *a(n-1) +24*(-27*n^2+47*n+10) *a(n-2) +48 *(27*n^2-47*n-10) *a(n-3) -12*(3*n-13)*(3*n-5)*(23*n^2-116*n+60) *a(n-4)=0. - R. J. Mathar, Jul 22 2022
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MAPLE
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a := proc(n) if n mod 2 = 0 then 4^(n/2)*binomial((3*(n/2)-1)/2, n/2)/2/(n/2+1) else 2*4^((n-1)/2)*binomial((3*((n-1)/2)-1)/2, (n-1)/2)/2/((n-1)/2+1) fi end; seq(a(n), n=1..30);
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MATHEMATICA
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a[n_] := If[EvenQ[n], 2^n Binomial[(3n-2)/4, n/2]/(n+2), 2^n Binomial[ (3n-5)/4, (n-1)/2]/(n+1)];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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