Search: a112944 -id:a112944
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A112945
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Number of unrooted regular odd-valent planar maps with 4 vertices; maps are considered up to orientation-preserving homeomorphisms and the vertices are of valency 2n+1.
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+10
3
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0, 6, 468, 80600, 16016560, 3360790440, 728936019504, 161858688461184, 36580777518027600, 8382066029146609800, 1941971956789550319920, 454006489072843947528288, 106944132919124515725427808
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (n/6)*binomial(2n, n)^4+(n/2)*binomial(2n, n)^2+(2/3)*delta(3|(2n+1))* binomial(2*floor(n/3), floor(n/3))*binomial(2n, n) where delta(3|(2n+1))=1 if 3|(2n+1) and =0 otherwise.
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MATHEMATICA
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a[n_] := (n/6) Binomial[2n, n]^4 + (n/2) Binomial[2n, n]^2 + (2/3) Boole[ Divisible[2n+1, 3]] Binomial[2 Floor[n/3], Floor[n/3]] Binomial[2n, n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A113181
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Number of unrooted two-vertex (or, dually, two-face) regular planar maps of even valency 2n considered up to orientation-preserving homeomorphism.
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+10
3
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1, 3, 14, 95, 859, 9130, 106039, 1297295, 16428300, 213388961, 2827645453, 38086408002, 520062618300, 7184570776213, 100256059855188, 1411319038583375, 20021022607979629, 285965560309310708, 4109498933510809561, 59380204746202961953, 862266486434574492404
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = binomial(2n,n)/4 + 1/(4n) Sum_{k|2n} phi(k) binomial((2n/k)-1),floor(n/k))^2 where phi(k) is the Euler function A000010.
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EXAMPLE
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There exist 3 planar maps with two 4-valent vertices: a map with four parallel edges and two different maps with two parallel edges and one loop in each vertex. Therefore a(2)=3.
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MATHEMATICA
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a[n_] := Binomial[2n, n]/4 + (1/(4n)) Sum[EulerPhi[k] Binomial[2n/k - 1, Floor[n/k]]^2, {k, Divisors[2n]}];
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PROG
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(PARI) a(n) = binomial(2*n, n)/4 + sumdiv(2*n, k, eulerphi(k)* binomial(2*n/k-1, (n\k))^2)/(4*n); \\ Michel Marcus, Oct 14 2015
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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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A113182
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Number of unrooted two-vertex (or, dually, two-face) regular planar maps of valency n considered up to orientation-preserving homeomorphism.
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+10
3
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1, 1, 2, 3, 7, 14, 39, 95, 308, 859, 3013, 9130, 33300, 106039, 394340, 1297295, 4878109, 16428300, 62232321, 213388961, 812825244, 2827645453, 10818489817, 38086408002, 146250545528, 520062618300, 2003199281223, 7184570776213
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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LINKS
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MATHEMATICA
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a[n_] := If[OddQ[n], (1/(2n))(Sum[EulerPhi[d] Binomial[2 Floor[(n-1)/(2d)], Floor[(n-1)/(2d)]]^2, {d, Divisors[n]}] + n Binomial[n-1, (n-1)/2]), (1/4)((2 Sum[EulerPhi[d] Binomial[n/d-1, Floor[n/(2d)]]^2, {d, Divisors[ n]}])/n + Binomial[n, n/2])];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A112948
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Number of unrooted 3-regular planar maps with 2n vertices, up to orientation-preserving isomorphisms.
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+10
2
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OFFSET
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1,1
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COMMENTS
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A 3-regular map is a regular map with valency 3.
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LINKS
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EXAMPLE
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There exist 2 planar maps with two 3-valent vertices: a map with three parallel edges and a map with one loop in each vertex and a link connecting the vertices. Therefore a(1)=2.
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CROSSREFS
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3-regular maps on the torus: A292408.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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