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.

A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368
Offset: 0

Views

Author

Vladeta Jovovic, Jun 23 2007

Keywords

Examples

			Triangle begins:
  1;
  1;
  1,  1;
  1,  3,  3;
  1,  6, 15,  16,   3;
  1, 10, 45, 110, 140, 60, 10;
  ...

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Row sums give A047864,
Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
The unlabeled version is A297877.
Cf. A052296, A033995.

Programs

  • Mathematica
    nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022

Formula

E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).

A053527 Number of bipartite graphs with 4 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 3, 140, 1125, 5355, 19075, 56133, 143955, 332475, 706860, 1404975, 2640638, 4733820, 8149050, 13543390, 21825450, 34227018, 52388985, 78463350, 115233195, 166252625, 236008773, 330108075, 455489125, 620664525, 835994250
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Column k=4 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges).

Programs

  • GAP
    List([0..30], n-> Binomial(n,4)*(n+2)*(n^3-5*n-36)/16 ) # G. C. Greubel, May 15 2019
  • Magma
    [(n^5-4*n^4-n^3+16*n^2-12*n)*(n^3-5*n-36)/384: n in [0..30]]; // Vincenzo Librandi, May 08 2012
  • Mathematica
    CoefficientList[Series[x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9, {x,0, 30}], x] (* Vincenzo Librandi, May 08 2012 *)
  • PARI
    {a(n) = binomial(n,4)*(n+2)*(n^3-5*n-36)/16}; \\ G. C. Greubel, May 15 2019
  • Sage
    [binomial(n,4)*(n+2)*(n^3-5*n-36)/16 for n in (0..30)] # G. C. Greubel, May 15 2019

Formula

a(n) = (n-3)*(n-2)*(n-1)*n*(n+2)*(n^3-5*n-36)/384.
G.f.: x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9. - Colin Barker, May 08 2012
E.g.f.: x^4*(48 + 400*x + 176*x^2 + 24*x^3 + x^4)*exp(x)/384. - G. C. Greubel, May 15 2019

A245796 T(n,k) is the number of labeled graphs of n vertices and k edges that have endpoints, where an endpoint is a vertex with degree 1.

Original entry on oeis.org

0, 1, 3, 3, 6, 15, 16, 12, 10, 45, 110, 195, 210, 120, 20, 15, 105, 435, 1320, 2841, 4410, 4845, 3360, 1350, 300, 30, 21, 210, 1295, 5880, 19887, 51954, 106785, 171360, 208565, 186375, 120855, 56805, 19110, 4410, 630, 42
Offset: 1

Views

Author

Chai Wah Wu, Aug 01 2014

Keywords

Comments

The length of the rows are 1,1,2,4,7,11,16,22,...: (1+(n-1)*(n-2)/2) = A152947(n).
T(n,k) = 0 if k > (n-1)*(n-2)/2 + 1.
Let j = (n-1)*(n-2)/2. For i >=0, n >= 4+i, T(n,j-i+1) = n*(n-1)*binomial(j,i).
For k <= 3, T(n,k) is equal to the number of labeled bipartite graphs with n vertices and k edges. In particular, T(n,1) = A000217(n-1), T(n,2) = A050534(n) and T(n,3) = A053526(n).

Examples

			Triangle starts:
..0
..1
..3......3
..6.....15.....16.....12
.10.....45....110....195....210....120.....20
.15....105....435...1320...2841...4410...4845...3360...1350....300.....30
...

Links

Crossrefs

Sum of n-th row is A245797(n).
Cf. A000217, A050534, A053526, A152947.

A053528 Number of bipartite graphs with 5 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 60, 1701, 14952, 81228, 331884, 1116675, 3256407, 8500734, 20306286, 45093048, 94189095, 186736368, 353904096, 644842674, 1134910242, 1936817820, 3215467584, 5207403663, 8245956642, 12793342716, 19481177100, 29161079805, 42967291185, 62393475690
Offset: 0

Views

Author

N. J. A. Sloane, Jan 16 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Links

Crossrefs

Column k=5 of A117279.
Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges), A053527 (4 edges).

Programs

  • GAP
    List([0..30], n-> Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32) # G. C. Greubel, May 15 2019
  • Magma
    [Binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32: n in [0..30]]; // G. C. Greubel, May 15 2019
  • Mathematica
    Table[Binomial[n,5]*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32, {n,0,30}] (* G. C. Greubel, May 15 2019 *)
  • PARI
    {a(n) = binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32}; \\ G. C. Greubel, May 15 2019
  • Sage
    [binomial(n,5)*(n^5 +5*n^4 +5*n^3 -85*n^2 -374*n -960)/32 for n in (0..30)] # G. C. Greubel, May 15 2019

Formula

a(n) = (n-4)*(n-3)*(n-2)*(n-1)*n*(n^5 + 5*n^4 + 5*n^3 - 85*n^2 - 374*n - 960)/3840.
G.f.: x^5*(60+1041*x-459*x^2+411*x^3-129*x^4+21*x^5)/(1-x)^11. - Colin Barker, May 08 2012
E.g.f.: x^5*(1920 + 7152*x + 3280*x^2 + 560*x^3 + 40*x^4 + x^5)*exp(x)/3840. - G. C. Greubel, May 15 2019
Showing 1-4 of 4 results.

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

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