A125208 -id:A125208 - cp's OEIS frontend

A125209 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^(n(n-1)/2-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.

1, 1, 1, -1, 3, 0, 1, 2, -6, 3, 4, 0, 0, 1, -6, 20, -15, -10, 10, 0, 5, 0, 0, 0, 1, 24, -90, 90, 25, -60, 0, -5, 15, 0, 0, 6, 0, 0, 0, 0, 1, -120, 504, -630, 0, 420, -105, 0, -105, 0, 35, -21, 21, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 720, -3360, 5040, -1176, -3150, 1680, 140, 560, -210, -280, 112

Offset: 1

Max Alekseyev, Jan 09 2007

Row-reversed version of A125208, see A125208 for further details.

			Triangle begins:
   1;
   1,  1;
  -1,  3,   0,   1;
   2, -6,   3,   4,  0, 0, 1;
  -6, 20, -15, -10, 10, 0, 5, 0, 0, 0, 1;
  ...

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

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

Max Alekseyev, Jan 09 2007

			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.

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

{ 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)))) }

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

Max Alekseyev, Mar 09 2015

G. C. Greubel, Table of n, a(n) for n = 1..30
Mathoverflow, Probability of a graph procedure.

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

[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

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 *)

{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 */

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

cp's OEIS Frontend

A125209 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^(n(n-1)/2-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.

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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.

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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).

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cp's OEIS Frontend

A125209 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^(n*(n-1)/2-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.

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Author

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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.

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Examples

Crossrefs

Programs

PARI

Formula

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).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A125209 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^(n(n-1)/2-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.