cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Andrew Howroyd, Nov 20 2020

Keywords

Crossrefs

Cf. A050535, A076864, A339036, A339038, A339042, A339043, A339044, A339065.

Programs

  • Mathematica
    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 *)
  • PARI
    \\ 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, []))}

Formula

G.f.: x*f(x)/g(x) where f(x) is the g.f. of A339065 and g(x) is the g.f. of A050535.

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

Views

Author

Andrew Howroyd, Nov 20 2020

Keywords

Crossrefs

Cf. A050535, A076864, A339036, A339037, A339038, A339040, A339043, A339065.

Programs

  • Mathematica
    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 *)
  • PARI
    \\ 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)}

Formula

G.f.: f(x) - g(x)^2 where x*f(x) is the g.f. of A339037 and g(x) is the g.f. of A339036.

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

Views

Author

Andrew Howroyd, Nov 20 2020

Keywords

Crossrefs

Cf. A050535, A076864, A339036, A339037, A339038, A339041, A339042, A339065.

Programs

  • Mathematica
    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 *)
  • PARI
    \\ 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}

Formula

G.f: f(g) - (g(x)^2 + g(x^2))/2 where x*f(x) is the g.f. of A339038 and g(x) is the g.f. of A339036.

A339038 Number of unlabeled connected loopless multigraphs with n edges rooted at one unoriented edge.

Original entry on oeis.org

1, 2, 7, 23, 88, 339, 1396, 5915, 26080, 118539, 555678, 2678458, 13262193, 67353325, 350493424, 1866989802, 10171394388, 56631507822, 322011612423, 1868702977253, 11061267210030, 66745602611831, 410360493588788, 2569318971123439, 16374787277199728, 106180292431149021
Offset: 1

Views

Author

Andrew Howroyd, Nov 20 2020

Keywords

Crossrefs

Cf. A050535, A076864, A303832, A339036, A339037, A339042, A339043, A339065, A339066.

Programs

  • Mathematica
    seq[n_] := (G[2n, x + O[x]^n, {1, 1}] + G[2n, x + O[x]^n, {2}])/G[2n, x + O[x]^n, {}] // CoefficientList[#/2, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)
  • PARI
    \\ 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, [2]))/G(2*n, x+A, []))/2}

Formula

G.f.: x*f(x)/g(x) where f(x) is the g.f. of A339066 and g(x) is the g.f. of A050535.

A339039 Number of unlabeled connected simple graphs with n edges rooted at one distinguished vertex.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 114, 367, 1248, 4446, 16526, 63914, 256642, 1067388, 4590201, 20376849, 93240065, 439190047, 2126970482, 10579017047, 53983000003, 282345671127, 1512273916781, 8287870474339, 46438619162441, 265840311066579
Offset: 0

Views

Author

Andrew Howroyd, Nov 20 2020

Keywords

Crossrefs

Cf. A000664, A002905, A053419, A126100, A303832, A339036, A339040, A339041, A339044, A339063.

Programs

  • PARI
    \\ See A339063 for G.
seq(n)={my(A=O(x*x^n)); Vec(G(2*n, x+A, [1])/G(2*n, x+A, []))}

Formula

G.f.: f(x)/g(x) where f(x) is the g.f. of A053419 and g(x) is the g.f. of A000664.

A339045 Number of connected loopless multigraphs with n edges rooted at two noninterchangeable vertices whose removal leaves a connected graph.

Original entry on oeis.org

1, 1, 4, 16, 69, 307, 1433, 6903, 34337, 175457, 919525, 4931233, 27023894, 151142376, 861880778, 5006906170, 29611120248, 178175786593, 1090266839041, 6781364484106, 42858210422338, 275127506187149, 1793418517202096, 11867326044069470, 79695273536227647
Offset: 1

Views

Author

Andrew Howroyd, Nov 25 2020

Keywords

Crossrefs

Cf. A338999, A339036, A339037, A339065.

Programs

  • PARI
    \\ 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); InvEulerT(Vec(-1+G(2*n, x+A, [1,1])/(g*gr^2)))}

Formula

1/(Product_{k>=1} (1 - x^k)^a(k)) = f(x)/g(x)^2 where x*f(x) is the g.f. of A339037 and g(x) is the g.f. of A339036.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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