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.

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.

A144209 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 4.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 15, 60, 1, 0, 0, 0, 45, 360, 1080, 1, 0, 0, 0, 105, 1260, 7560, 20580, 1, 0, 0, 0, 210, 3360, 30240, 164640, 430080, 1, 0, 0, 0, 378, 7560, 90720, 740880, 3873240, 9920232, 1, 0, 0, 0, 630, 15120, 226800, 2469600, 19367460, 99406440, 252000000
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2008

Keywords

Examples

			T(5,4) = 15 = 5*3, because there are 5 possibilities for a single node and T(4,4) = 3:
.1-2. .1-2. .1.2.
.|.|. ..X.. .|X|.
.3-4. .3-4. .3.4.
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 0, 0, 0;
1, 0, 0, 0,  3;
1, 0, 0, 0, 15, 60;

Links

Crossrefs

Columns 0, 1+2+3, 4 give: A000012, A000004, A050534.
Main diagonal gives A065889.
Row sums give A144210.
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, 3*Binomial[n-1, 3]*n^(n-4), True, T[n-1, k] + Sum[Binomial[n-1, j]*T[j+1, j+1]*T[n-1-j, k-j-1], {j, 3, k-1}]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2014, translated from Maple *)

Formula

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

A065888 a(n) = number of endofunctions on [n] with a 4-cycle a->b->c->d->a and for any x in [n], some iterate f^k(x) = a.

Original entry on oeis.org

6, 120, 2160, 41160, 860160, 19840464, 504000000, 14030763120, 425681879040, 13997939172360, 496360987938816, 18891066796875000, 768426686420090880, 33279382190563948320, 1529238539734890577920, 74326797938267012471904
Offset: 4

Views

Author

Len Smiley, Nov 27 2001

Keywords

Examples

			a(4) = 6 : 3 [choices of 1's opposite in cycle] * 2 [choices of 1's image]

Links

Crossrefs

Cf. A000169 (1-cycle), A053506 (2-cycle), A065513 (3-cycle), A065889 (= A065888/2: underlying simple graphs).

Programs

  • Mathematica
    Rest[Rest[Rest[Rest[CoefficientList[Series[(LambertW[-x])^4/4, {x, 0, 20}], x]* Range[0, 20]!]]]] (* Vaclav Kotesovec, Oct 05 2013 *)
Table[(n-1)(n-2)(n-3)n^(n-4),{n,4,20}] (* Harvey P. Dale, Dec 04 2015 *)

Formula

E.g.f.: T^4/4 where T = T(x) is Euler's tree function (see A000169).
a(n) = (n-1)*(n-2)*(n-3)*n^(n-4). - Vaclav Kotesovec, Oct 05 2013

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.
Showing 1-4 of 4 results.

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