A125210 -id:A125210 - cp's OEIS frontend

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.

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

Max Alekseyev, Jan 09 2007

			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.

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

{ 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)))) }

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

A127258 Irregular triangle read by rows: B(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that SUM B(n,k)q^(n(n-1)/2-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, -1, 2, 1, 0, -3, 3, 2, -6, 3, 4, 0, -6, 4, -6, 40, -105, 130, -60, -18, 15, 10, 0, -10, 5, -24, 270, -1350, 3925, -7260, 8712, -6485, 2445, 60, -330, -18, 45, 20, 0, -15, 6, 120, -2016, 15750, -75810, 250950, -603435, 1084104, -1471305, 1502550, -1128820, 589281, -182721

Offset: 1

Max Alekseyev, Jan 09 2007

Row-reversed version of A125210, see A125210 for further details.

			Triangle begins:
   1;
  -1,  2;
   1,  0,   -3,   3;
   2, -6,    3,   4,   0,  -6,  4;
  -6, 40, -105, 130, -60, -18, 15, 10, 0, -10, 5;
   ...

Cf. A125210 (row-reversed version), A125209 (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(Vec((1-q)^(n*(n-1)/2)*n!*polcoeff(B, n, x)))) }

cp's OEIS Frontend

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.

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A127258 Irregular triangle read by rows: B(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that SUM B(n,k)q^(n(n-1)/2-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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PARI

cp's OEIS Frontend

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.

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PARI

Formula

A127258 Irregular triangle read by rows: B(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that SUM B(n,k)*q^(n*(n-1)/2-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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Author

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Comments

Examples

Crossrefs

Programs

PARI

A127258 Irregular triangle read by rows: B(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that SUM B(n,k)q^(n(n-1)/2-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.