A339036
Number of unlabeled connected loopless multigraphs with n edges rooted at one distinguished vertex.
Original entry on oeis.org
1, 1, 3, 9, 30, 104, 390, 1518, 6208, 26372, 116221, 529341, 2487054, 12027502, 59778867, 304916272, 1594273763, 8535706749, 46753269749, 261771468438, 1497087288210, 8739579074131, 52045067963540, 315980654042243, 1954770128712348, 12315770916526091
Offset: 0
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seq[n_] := G[2n, x+O[x]^n, {1}]/G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}
A339037
Number of unlabeled connected loopless multigraphs with n edges rooted at one oriented edge.
Original entry on oeis.org
1, 3, 11, 41, 160, 641, 2672, 11479, 50938, 232830, 1095151, 5292990, 26257328, 133548307, 695752146, 3709509938, 20224607541, 112675185837, 641016837378, 3721624588590, 22037618432547, 133023405207408, 818085097509494, 5123460267381837, 32660335570381961, 211825198708110059
Offset: 1
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seq[n_] := Module[{A = O[x]^n}, G[2n, x+A, {1, 1}]/G[2n, x+A, {}] // CoefficientList[#, x]&]; (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
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\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1,1])/G(2*n, x+A, []))}
A339042
Number of unlabeled connected loopless multigraphs with n edges rooted at two noninterchangeable vertices.
Original entry on oeis.org
1, 4, 17, 73, 319, 1423, 6499, 30374, 145302, 711177, 3559690, 18212192, 95193547, 508083746, 2767835600, 15382476029, 87177582535, 503610832756, 2964300557548, 17771210411578, 108471258414870, 673836620069035, 4258727230198033, 27373904651169023, 178885471934461869
Offset: 1
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seq[n_] := Module[{g}, g = G[2n, x+O[x]^n, {}]; G[2n, x+O[x]^n, {1, 1}]/g - (G[2n, x+O[x]^n, {1}]/g)^2 // CoefficientList[#, x]& // Rest];
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)
-
\\ See A339065 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, [])); Vec(G(2*n, x+A, [1, 1])/g - (G(2*n, x+A, [1])/g)^2)}
A339043
Number of unlabeled connected loopless multigraphs with n edges rooted at two indistinguishable vertices.
Original entry on oeis.org
1, 3, 11, 43, 178, 767, 3425, 15783, 74775, 363639, 1811808, 9239430, 48175945, 256658465, 1396152633, 7750325528, 43882706171, 253308596926, 1490040961732, 8928063141435, 54469529215562, 338236254603888, 2136952452531537, 13731571816349732, 89710429044324926
Offset: 1
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seq[n_] := Module[{g, gr}, g = G[2n, x+O[x]^n, {}]; gr = G[2n, x+O[x]^n, {1}]/g; G[2n, x+O[x]^n, {1, 1}]/g - gr^2 + G[2n, x+O[x]^n, {2}]/g - (Normal[gr] /. x -> x^2) // CoefficientList[#/2, x]& // Rest];
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
-
\\ See A339065 for G.
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A, []), gr=G(2*n, x+A, [1])/g); Vec(G(2*n, x+A, [1, 1])/g - gr^2 + G(2*n, x+A, [2])/g - subst(gr, x, x^2))/2}
A338999
Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.
Original entry on oeis.org
1, 1, 3, 11, 43, 180, 804, 3763, 18331, 92330, 478795, 2547885, 13880832, 77284220, 439146427, 2543931619, 15010717722, 90154755356, 550817917537, 3421683388385, 21601986281226, 138548772267326, 902439162209914, 5967669851051612, 40053432076016812
Offset: 1
The a(3) = 3 CDE-descendants of A-Z with 3 edges are
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A A A
( ) / /
o o - o o - o
| / \
Z Z Z
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DCC DD DE
.
- Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.
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\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g, u=InvEulerT(Vec(-1+G(2*n, x+A,[1,1])/(g*gr^2))), t=InvEulerT(Vec(-1+G(2*n, x+A,[2])/(g*subst(gr,x,x^2)))), v=vector(n)); for(n=1, #v, v[n]=(u[n]+t[n]-if(n%2==0,u[n/2]-v[n/2]))/2); v} \\ Andrew Howroyd, Nov 20 2020
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