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.

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A098909 Triangle T(n,k) of numbers of connected (unicyclic) graphs with unique cycle of length k (3<=k<=n), on n labeled nodes.

Original entry on oeis.org

1, 12, 3, 150, 60, 12, 2160, 1080, 360, 60, 36015, 20580, 8820, 2520, 360, 688128, 430080, 215040, 80640, 20160, 2520, 14880348, 9920232, 5511240, 2449440, 816480, 181440, 20160, 360000000, 252000000, 151200000, 75600000, 30240000, 9072000
Offset: 3

Views

Author

Vladeta Jovovic, Oct 15 2004

Keywords

Examples

			Triangle begins as:
      1;
     12,     3;
    150,    60,   12;
   2160,  1080,  360,   60;
  36015, 20580, 8820, 2520, 360;
  ...

Links

Crossrefs

Row sums: A057500, columns: A053507, A065889.

Programs

  • GAP
    Flat(List([3..12], n-> List([3..n], k-> Factorial(k)*Binomial(n,k) *n^(n-k-1)/2 ))); # G. C. Greubel, May 16 2019
  • Magma
    [[Factorial(k)*Binomial(n,k)*n^(n-k-1)/2: k in [3..n]]: n in [3..12]]; // G. C. Greubel, May 16 2019
  • Mathematica
    f[list_] := Select[list, #>0&]; t = Sum[n^(n-1)x^n/n!, {n, 1, 20}]; Map[f,Drop[Transpose[Table[Range[0,8]! CoefficientList[Series[t^n/(2n), {x, 0, 8}], x], {n, 3, 8}]], 3]] (* Geoffrey Critzer, Oct 23 2011 *)
Table[k!*Binomial[n,k]*n^(n-k-1)/2, {n,3,12}, {k,3,n}]//Flatten (* G. C. Greubel, May 16 2019 *)
  • PARI
    {T(n,k) = k!*binomial(n,k)*n^(n-k-1)/2 }; \\ G. C. Greubel, May 16 2019
  • Sage
    [[factorial(k)*binomial(n,k)*n^(n-k-1)/2 for k in (3..n)] for n in (3..12)] # G. C. Greubel, May 16 2019

Formula

T(n, k) = (n-1)!*n^(n-k)/(2*(n-k)!).
E.g.f.: -(2*log(1+x*LambertW(-y))-2*x*LambertW(-y)+x^2*LambertW(-y)^2)/4.

A144207 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 3.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 12, 1, 0, 0, 10, 60, 150, 1, 0, 0, 20, 180, 900, 2160, 1, 0, 0, 35, 420, 3150, 15180, 36015, 1, 0, 0, 56, 840, 8400, 60750, 291060, 688128, 1, 0, 0, 84, 1512, 18900, 182270, 1311240, 6300672, 14880348, 1, 0, 0, 120, 2520, 37800
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2008

Keywords

Examples

			T(5,4) = 60 = 5*12, because there are 5 possibilities for a single node and T(4,4) = 12:
.1-2. .1-2. .1-2. .1.2. .1.2. .1-2. .1.2. .1.2. .1-2. .1-2. .1-2. .1-2.
.|X.. .|/|. .|/.. ..X|. .|/|. ../|. .|X.. .|\|. .|\.. ..X|. .|\|. ..\|.
.3.4. .3.4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3.4. .3.4. .3-4.
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 0, 0, 1;
1, 0, 0, 4, 12;
1, 0, 0, 10, 60, 150;

Links

Crossrefs

Columns 0, 1+2, 3, 4 give: A000012, A000004, A000292, A033486 or A112415. Diagonal gives: A053507. Row sums give: A144208. Cf. A007318.

Programs

  • Maple
    T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n
  • Mathematica
    t[n_, k_] := t[n, k] = Which[k == 0, 1, k < 0 || n < k, 0, k == n, Binomial[n-1, 2]*n^(n-3), True, t[n-1, k] + Sum[Binomial[n-1, j]*t[j+1, j+1]*t[n-1-j, k-j-1], {j, 2, k-1}]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Formula

T(n,0) = 1, T(n,k) = 0 if k<0 or n

A144208 Number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 3; also row sums of A144207.

Original entry on oeis.org

1, 1, 1, 2, 17, 221, 3261, 54801, 1049235, 22695027, 548904831, 14701691121, 432342705351, 13856514927207, 480891887472585, 17971038945463101, 719613541474095591, 30743125693699501431, 1395902175504288127695
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2008

Keywords

Examples

			a(3) = 2, because there are 2 simple graphs on 3 labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 3:
.1.2. .1-2.
..... .|/..
.3... .3...

Links

Crossrefs

Row sums of triangle A144207. A column of A144212. Cf. A053507, A007318.

Programs

  • Maple
    T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n add(T(n,k), k=0..n): seq(a(n), n=0..23);
  • Mathematica
    T[n_, k_] := T[n, k] = Which[k == 0, 1, k<0 || nJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} A144207(n,k).
a(n) ~ c * n^(n-1), where c = 0.762590842281789937101466... . - Vaclav Kotesovec, Sep 10 2014

A144212 Triangle T(n,k), n>=3, 3<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length k.

Original entry on oeis.org

2, 17, 4, 221, 76, 13, 3261, 1486, 433, 61, 54801, 29506, 11593, 2941, 361, 1049235, 628531, 296353, 102481, 23041, 2521, 22695027, 14633011, 7795873, 3270961, 1010881, 204121, 20161, 548904831, 373486051, 217126225, 104038201, 39355201
Offset: 3

Views

Author

Alois P. Heinz, Sep 14 2008

Keywords

Examples

			T(4,4) = 4, because there are 4 simple graphs on 4 labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 4:
.1.2. .1-2. .1-2. .1.2.
..... .|.|. ..X.. .|X|.
.3.4. .3-4. .3-4. .3.4.
Triangle begins:
        2;
       17,     4;
      221,    76,    13;
     3261,  1486,   433,   61;
    54801, 29506, 11593, 2941, 361;

Links

Crossrefs

Columns k=3, 4 give: A144208, A144210. Diagonal gives: A139149. Cf. A053507, A065889, A098909, A144207, A144209, A007318, A000142.

Programs

  • Maple
    B:= proc(n,c,k) option remember; if c=0 then 1 elif c<0 or n add(B(n,c,k), c=0..n): seq(seq(T(n,k), k=3..n), n=3..11);
  • Mathematica
    B[n_, c_, k_] := B[n, c, k] = Which[c == 0, 1, c<0 || nJean-François Alcover, Jan 21 2014, translated from Alois P. Heinz's Maple code *)

Formula

See program.
Previous Showing 11-14 of 14 results.

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