Search: a118094 -id:a118094
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A090371
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Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.
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+10
10
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1, 3, 6, 20, 60, 291, 1310, 6975, 37746, 215602, 1262874, 7611156, 46814132, 293447817, 1868710728, 12068905911, 78913940784, 521709872895, 3483289035186, 23464708686960, 159346213738020, 1090073011199451, 7507285094455566, 52021636161126702
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OFFSET
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1,2
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COMMENTS
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a(n) is also the number of unrooted planar hypermaps with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets, Apr 13 2006
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LINKS
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EXAMPLE
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The 3 Eulerian maps with 2 edges are the digon and two figure eight graphs ("8") in which both loops are colored, resp., black or white.
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MAPLE
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local s, d;
if n=0 then
1 ;
else
s := -2^n*binomial(2*n, n);
for d in numtheory[divisors](n) do
s := s+ numtheory[phi](n/d)*2^d*binomial(2*d, d)
od;
3/(2*n)*(2^n*binomial(2*n, n)/((n+1)*(n+2))+s/2);
fi;
end proc:
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MATHEMATICA
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h0[n_] := 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)); a[n_] := (h0[n] + DivisorSum[n, If[#>1, EulerPhi[#]*Binomial[n/#+2, 2]*h0[n/#], 0]&])/n; Array[a, 30] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
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PROG
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(PARI) h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n; \\ Michel Marcus, Dec 11 2014
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CROSSREFS
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KEYWORD
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easy,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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A214819
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Number of genus 2 sensed hypermaps with n darts.
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+10
7
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0, 0, 0, 0, 4, 48, 708, 9807, 119436, 1355400, 14561360, 150429819
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OFFSET
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1,5
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A214820
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Number of genus 3 sensed hypermaps with n darts.
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+10
7
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0, 0, 0, 0, 0, 0, 30, 1155, 29910, 601364, 10260804, 156469887
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listen;
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OFFSET
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1,7
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A118093
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Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).
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+10
6
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1, 15, 165, 1611, 14805, 131307, 1138261, 9713835, 81968469, 685888171, 5702382933, 47168678571, 388580070741, 3190523226795, 26124382262613, 213415462218411, 1740019150443861, 14162920013474475, 115112250539595093, 934419385591442091, 7576722323539318101
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OFFSET
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3,2
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LINKS
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FORMULA
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Conjecture: +n*(5*n-17)*a(n) -15*(n-1)*(5*n-16)*a(n-1) +12*(20*n^2-103*n+140)*a(n-2) +32*(5*n-12)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 05 2018
G.f.: (1 - 7*x + 4*x^2 - (1 - 3*x)*sqrt(1 - 8*x))/(8*(1 + x)*(1 - 8*x)); equivalently, the g.f. can be rewritten as (y - 1)^3/(4*(y - 2)^2*(y + 1)), where y=G(2*x) with G the g.f. of A000108. - Gheorghe Coserea, Nov 06 2018
a(n) ~ 2^(3*n - 4) / 3 * (1 - 10/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Nov 06 2018
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MATHEMATICA
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Table[Sum[2^k (4^(n - 2 - k) - 1) Binomial[n+k, k] / 3, {k, 0, n-3}], {n, 3, 25}] (* Vincenzo Librandi, Sep 16 2018 *)
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PROG
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(PARI) a(n) = sum(k=0, n-3, 2^k*(4^(n-2-k)-1)*binomial(n+k, k))/3; \\ Michel Marcus, Dec 11 2014
(PARI)
seq(N) = {
my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
Vec((y - 1)^3/(4*(y - 2)^2*(y + 1)));
};
(Magma) [&+[(2^k*(4^(n-2-k)-1)*Binomial(n+k, k))/3 : k in [0..n-3]]: n in [3..25]]; // Vincenzo Librandi, Sep 16 2018
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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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A214821
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Number of genus 0 unsensed hypermaps with n darts.
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+10
5
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1, 3, 6, 20, 57, 240, 954, 4566, 22641, 121823, 683307, 4004055
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OFFSET
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1,2
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A214823
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Number of genus 2 unsensed hypermaps with n darts.
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+10
5
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0, 0, 0, 0, 4, 39, 456, 5554, 63378, 698568, 7391499, 75807708
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OFFSET
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1,5
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A215017
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Number of genus 3 unsensed hypermaps with n darts.
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+10
5
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0, 0, 0, 0, 0, 0, 25, 678, 15867, 307880, 5180472, 78573507
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OFFSET
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1,7
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A215018
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Number of unsensed hypermaps with n darts and any genus.
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+10
5
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1, 3, 7, 26, 91, 490, 2785, 20434, 171579, 1671193, 18192737, 218487504
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OFFSET
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1,2
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A214822
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Number of genus 1 unsensed hypermaps with n darts.
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+10
0
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0, 0, 1, 6, 30, 211, 1350, 9636, 69169, 513012, 3843024, 29107494
(list;
graph;
refs;
listen;
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OFFSET
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1,4
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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