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 1-5 of 5 results.

A125207 Total number of connected components in all subgraphs obtained from the complete labeled graph K_n by removing zero or more edges.

Original entry on oeis.org

1, 3, 13, 98, 1398, 39956, 2354240, 286394544, 71225744048, 35884971729760, 36419817759267072, 74221711070826087424, 303193538300703211111936, 2480118087478081928075065344, 40601989279034990139321984265216, 1329877330680067685563700135615633408
Offset: 1

Views

Author

Max Alekseyev, Nov 23 2006

Keywords

Comments

a(n)/A006125(n) is the expected number of connected components in a simple labeled graph on n vertices. - Geoffrey Critzer, May 09 2011

Examples

			For n=2, we have two graph on two vertices: complete and empty, the former has one connected component while the latter has two connected components. The total number of connected components is 3, which is a(2).

Links

Crossrefs

Cf. A001187, A125205, A125206, A143543.

Programs

  • Mathematica
    f[list_]:= Total[Table[i list[[i]],{i,1,Length[list]}]];
a= Sum[2^Binomial[n,2] x^n/n!,{n,0,20}];
Map[f, Transpose[Table[Rest[Range[0, 20]! CoefficientList[Series[Log[a]^k/k!, {x, 0, 20}],x]], {k, 1, 20}]]] (* Geoffrey Critzer, May 09 2011 *)
  • PARI
    G=sum(n=0,30,2^(n*(n-1)/2)*x^n/n!) + O(x^31); v=Vec(G*log(G)); for(i=1,length(v),v[i]*=i!); print(v)

Formula

E.g.f.: (F(x)-1)*exp(F(x)-1) = G(x)*log(G(x)) where G(x) = Sum_{n>=0} 2^(n(n-1)/2) * x^n/n! and F(x) = 1+log(G(x)) is the e.g.f. of A001187.
a(n) = Sum_{k=1..n} k * A143543(n,k). - Alois P. Heinz, Feb 02 2024

A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).

Original entry on oeis.org

1, 2, 1, 3, 6, 3, 1, 4, 18, 30, 24, 15, 6, 1, 5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1, 6, 75, 420, 1385, 3015, 4800, 6365, 7170, 6705, 5065, 3009, 1365, 455, 105, 15, 1, 7, 126, 1050, 5355, 18690, 47880, 96796, 166890, 251370, 329945, 373947, 362292, 297115
Offset: 1

Views

Author

Max Alekseyev, Nov 23 2006

Keywords

Examples

			Triangle begins:
  1;
  2,  1;
  3,  6,   3,   1;
  4, 18,  30,  24,  15,   6,   1;
  5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1;
  ...
T(3,1) = 6 since there are three different subgraphs of K_3 with one edge and each subgraph has two connected components.

Crossrefs

Cf. A062734.
Cf. A125206 (row-reversed version), A125207 (row sums).

Programs

  • PARI
    { G=sum(n=0,6,(1+y)^(n*(n-1)/2)*x^n/n!); K=G*log(G); for(n=1,6,print(Vecrev(n!*polcoeff(K,n,x)))) }

Formula

G.f.: Sum_{n,k} T(n,k)*x^n/n!*y^k=(F(x,y)-1)*exp(F(x,y)-1)=G(x,y)*log(G(x,y)) where G(x,y)=Sum_{n=0..oo} (1+y)^(n(n-1)/2)*x^n/n! and F(x,y)=1+log(G(x,y)) is g.f. of A062734.

A125208 Irregular triangle read by rows: T(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that Sum_k T(n,k)*p^k gives the expectation of the number of connected components after deleting every edge of the complete graph on n labeled vertices with probability p.

Original entry on oeis.org

1, 1, 1, 1, 0, 3, -1, 1, 0, 0, 4, 3, -6, 2, 1, 0, 0, 0, 5, 0, 10, -10, -15, 20, -6, 1, 0, 0, 0, 0, 6, 0, 0, 15, -5, 0, -60, 25, 90, -90, 24, 1, 0, 0, 0, 0, 0, 7, 0, 0, 0, 21, -21, 35, 0, -105, 0, -105, 420, 0, -630, 504, -120, 1, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 28, -28, 0, 56, 35, -168, 112, -280
Offset: 1

Views

Author

Max Alekseyev, Jan 09 2007

Keywords

Examples

			Triangle begins:
  1;
  1, 1;
  1, 0, 3, -1;
  1, 0, 0,  4, 3, -6,  2;
  1, 0, 0,  0, 5,  0, 10, -10, -15, 20, -6;
   ...
Sum_k T(3,k)*p^k = 1+3*p^2-p^3 is the expectation of the number of connected components in a complete graph on 3 labeled vertices where every edge is removed with probability p.

Crossrefs

Cf. A125205, A125206, A125209 (row-reversed version), A125210 (dual version).

Programs

  • PARI
    { H=sum(n=0,6,x^n/p^(n*(n-1)/2)/n!); A=H*log(H); for(n=1,6,print(Vecrev(p^(n*(n-1)/2)*n!*polcoeff(A,n,x)))) }

Formula

G.f.: Sum_{n,k} T(n,k)*x^n/(p^(n*(n-1)/2)*n!) = H(x,p)*exp(H(x,p)) where H(x,p)=Sum_{n=1..oo} x^n/(p^(n*(n-1)/2)*n!).
Sum_k T(n,k)*p^k = Sum_k A125205(n,k)*p^(n*(n-1)/2-k)*(1-p)^k

A125210 Irregular triangle read by rows: T(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that Sum_k T(n,k)*q^k gives the expectation of the number of connected components in a random graph on n labeled vertices where every edge is present with probability q.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 1, 4, -6, 0, 4, 3, -6, 2, 5, -10, 0, 10, 15, -18, -60, 130, -105, 40, -6, 6, -15, 0, 20, 45, -18, -330, 60, 2445, -6485, 8712, -7260, 3925, -1350, 270, -24, 7, -21, 0, 35, 105, 42, -980, -1950, 11760, 12355, -182721, 589281, -1128820, 1502550, -1471305
Offset: 1

Views

Author

Max Alekseyev, Jan 09 2007

Keywords

Examples

			Triangle begins:
  1;
  2,  -1;
  3,  -3,  0,  1;
  4,  -6,  0,  4,  3,  -6,   2;
  5, -10,  0, 10, 15, -18, -60, 130, -105, 40, -6;
...
Sum_k T(3,k)*q^k = 3-3*q+q^3 is the expectation of the number of connected components in a random graph on 3 labeled vertices where every edge is present with probability q.

Crossrefs

Cf. A125205, A125206, A127258 (row-reversed version), A125208 (dual version).

Programs

  • PARI
    { H=sum(n=0,6,x^n/(1-q)^(n*(n-1)/2)/n!); B=H*log(H); for(n=1,6,print(Vecrev((1-q)^(n*(n-1)/2)*n!*polcoeff(B,n,x)))) }

Formula

G.f.: Sum_{n,k} T(n,k)*x^n/((1-q)^(n*(n-1)/2)*n!) = H(x,1-q)*exp(H(x,1-q)) where H(x,p)=Sum_{n=1..oo} x^n/(p^(n*(n-1)/2)*n!).
Sum_k T(n,k)*q^k = Sum_k A125205(n,k)*(1-q)^(n*(n-1)/2-k)*q^k
Sum_k T(n,k)*q^k = Sum_k A125206(n,k)*q^(n*(n-1)/2-k)*(1-q)^k

A255886 Number of orderings of the edges of the labeled complete graph K_n such that the graph induced by the first k edges is connected for every k=1,2,...,binomial(n,2).

Original entry on oeis.org

1, 1, 6, 576, 2073600, 498161664000, 12385682950717440000, 45484508287062207627264000000, 33297304775599549535597153400913920000000, 6298496203530014357849150420174490961843322880000000000, 387030157006015555733158587399026951851936435957496524308480000000000000
Offset: 1

Views

Author

Max Alekseyev, Mar 09 2015

Keywords

Links

Crossrefs

Cf. A125205, A125206, A125207, A125208, A125209.

Programs

  • Magma
    [1] cat [Factorial(Binomial(n,2))*2^(n-2)*n/Binomial(2*n-2,n-1): n in [2..20]]; // G. C. Greubel, Aug 03 2018
  • Mathematica
    Join[{1}, Table[Binomial[n, 2]!*2^(n-2)*n/Binomial[2*n-2, n-1], {n, 2, 20}]] (* G. C. Greubel, Aug 03 2018 *)
  • PARI
    {a(n) = if( n<2, n>0, binomial(n, 2)! * 2^(n-2) * n / binomial(2*n-2, n-1))}; /* Michael Somos, Jul 23 2015 */

Formula

For n>1, a(n) = binomial(n,2)! * 2^(n-2) / A000108(n-1).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.