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 1-4 of 4 results.

A290716 Number of minimal dominating sets in the n-triangular (Johnson) graph.

Original entry on oeis.org

1, 1, 1, 3, 15, 35, 225, 1197, 6881, 45369, 327375, 2460755, 19925367, 171368067, 1551364997, 14763620445, 147405166785, 1538113071857, 16732908859599, 189413984297187, 2226589748578775, 27130592749003275, 342118450334269917, 4458168165784234253, 59952936723606219009
Offset: 0

Views

Author

Eric W. Weisstein, Aug 09 2017

Keywords

Comments

A minimal dominating set on the triangular graph corresponds either with a minimal edge cover on the complete graph minus one vertex or with a perfect matching on the complete graph. Perfect matchings on the complete graph exists only for even n. - Andrew Howroyd, Aug 13 2017
Also the number of maximal irredundant sets in the n-triangular graph. - Eric W. Weisstein, Dec 31 2017

Links

Crossrefs

Cf. A001147, A053530, A290847.

Programs

  • Mathematica
    b[n_]:=n! Sum[1/k! (Binomial[k, n - k] 2^(k - n) (-1)^k + Sum[Binomial[k, j] Sum[j^(i - j)/(i - j)! Binomial[k - j, n - i - k + j] 2^(i - j + k - n) (-1)^(k - j), {i, j, n - k + j}], {j, k}]), {k, n}]; Join[{1, 1}, Table[n b[n - 1] + If[Mod[n, 2] == 0, (n - 1)!!, 0], {n, 2, 20}]] (* Eric W. Weisstein, Aug 14 2017 *)
Range[0, 20]! CoefficientList[Series[Exp[x^2/2] + x Exp[x Exp[x] - (x + x^2/2)], {x, 0, 20}], x] (* Eric W. Weisstein, Apr 23 2018 *)
  • PARI
    \\ here b(n) is A053530, df(n) is (2*n-1)!! = A001147
b(n)=polcoeff(serlaplace(exp(-x-1/2*x^2+x*exp(x+O(x^(n+1))))),n,x);
df(n)=polcoeff(serlaplace((1-2*x+O(x^(n+1)))^(-1/2)),n,x);
a(n) = n*b(n-1) + if(n%2==0, df(n/2), 0); \\ Andrew Howroyd, Aug 13 2017
  • PARI
    seq(n)={Vec(serlaplace(exp(x^2/2 + O(x*x^n)) + x*exp(x*exp(x + O(x^n)) - (x+x^2/2))))} \\ Andrew Howroyd, Apr 21 2018

Formula

a(n) = n*A053530(n-1) for n odd, a(n) = (n-1)!! + n*A053530(n-1) for n even. - Andrew Howroyd, Aug 13 2017
E.g.f.: exp(x^2/2) + x*exp(x*exp(x) - (x+x^2/2)). - Andrew Howroyd, Apr 21 2018

Extensions

a(8)-a(24) from formula by Andrew Howroyd, Aug 13 2017
a(0)-a(1) prepended by Andrew Howroyd, Apr 21 2018

A303048 Number of total dominating sets in the n-triangular (Johnson) graph.

Original entry on oeis.org

0, 4, 54, 918, 31232, 2059624, 266734812, 68574627036, 35160753222400, 36021330363615408, 73782362964470935112, 302225854825997535378632, 2475866675779140063716682240, 40564755806137338166417907530592, 1329227401912999475682655581004557840
Offset: 2

Views

Author

Eric W. Weisstein, Apr 17 2018

Keywords

Links

Crossrefs

Cf. A006129, A290847, A298104, A303227, A304561.

Programs

  • Mathematica
    b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}]
Table[Sum[(-1)^k Binomial[n, 2 k] (2 k)!/(2^k k!) (b[n - 2 k] + (n - 2 k) b[n - 2 k - 1]), {k, 0, Floor[n/2]}], {n, 2, 20}]
  • PARI
    \\ here b(n) is A006129
b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
a(n)=sum(k=0, n\2, (-1)^k*binomial(n,2*k)*(2*k)!/(2^k*k!)*(b(n-2*k) + (n-2*k)*b(n-2*k-1))); \\ Andrew Howroyd, Apr 20 2018

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*(2*k-1)!!*A290847(n-2*k). - Andrew Howroyd, Apr 20 2018

Extensions

a(8)-a(16) from Andrew Howroyd, Apr 20 2018

A298104 Number of connected dominating sets in the n-triangular graph.

Original entry on oeis.org

1, 7, 54, 918, 31072, 2053184, 266478640, 68560228400, 35159451505536, 36021118923496320, 73782296097354062336, 302225812400352055040512, 2475866621867539032536216576, 40564755669895936890639118713856, 1329227401230786888692092742930946048
Offset: 2

Views

Author

Eric W. Weisstein, Jan 12 2018

Keywords

Links

Crossrefs

Cf. A001187, A290716, A290847.

Programs

  • Mathematica
    a[n_] := a[n] = 2^Binomial[n, 2] - Sum[k Binomial[n, k] 2^((n - k) (n - k - 1)/2) a[k], {k, n - 1}]/n; Join[{1}, Table[a[n] + n a[n - 1], {n, 3, 20}]] (* Eric W. Weisstein, Jan 15 2018 *)
  • PARI
    b(n)={serlaplace(-x + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))))}
{ my(n=20); Vec(b(n) + x*deriv(x*b(n))) } \\ Andrew Howroyd, Jan 14 2018

Formula

a(n) = A001187(n) + n*A001187(n-1) for n > 2. - Andrew Howroyd, Jan 14 2018

Extensions

a(8)-a(16) from Andrew Howroyd, Jan 14 2018

A323499 Number of minimum dominating sets in the n-triangular (Johnson) graph.

Original entry on oeis.org

1, 3, 15, 15, 195, 105, 2625, 945, 38745, 10395, 634095, 135135, 11486475, 2027025, 229053825, 34459425, 4996616625, 654729075, 118505962575, 13749310575, 3038597637075, 316234143225, 83802047954625, 7905853580625, 2474532170735625, 213458046676875
Offset: 2

Views

Author

Eric W. Weisstein, Jan 16 2019

Keywords

Links

Crossrefs

Cf. A290716, A290847, A298104, A303048, A303227, A304561.

Programs

  • PARI
    a(n)={my(m=(n+1)\2); ((2*m)!/(m!*2^m))*if(n%2, 1, 1 + n*(n/2-1))} \\ Andrew Howroyd, Sep 08 2019

Formula

a(n) = n!! for n odd.
a(n) = (n-1)!!*(1 + n*(n/2 - 1)) for n even. - Andrew Howroyd, Sep 08 2019

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 08 2019
Showing 1-4 of 4 results.

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