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 1-2 of 2 results.

A351589 Number of minimal edge covers in the n-cocktail party graph.

Original entry on oeis.org

0, 2, 74, 2228, 100494, 6014932, 453143662, 41921209920, 4639656895118, 603202689990836, 90714189165482310, 15583340701180474312, 3025677781064563172326, 658038493760685537784572, 159065982382639942877853134, 42449055613405195868802686816
Offset: 1

Views

Author

Eric W. Weisstein, Feb 14 2022

Keywords

Links

Crossrefs

Cf. A053530, A297029, A297474, A351588.

Programs

  • PARI
    a(n)={my(x=x+O(x^(2*n+1)), p=exp(-x - x^2/2 + x*exp(x)), q=2*exp(x) - 1); sum(k=0, n, (-1)^(n-k)*binomial(n,k)*(2*k)!*polcoef(q^(n-k)*p, 2*k))} \\ Andrew Howroyd, Feb 21 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (2*k)! * [x^(2*k)] B(n-k,x), where B(k,x) = (2*exp(x) - 1)^k * exp(-x - x^2/2 + x*exp(x)). - Andrew Howroyd, Feb 21 2022

Extensions

Terms a(5) and beyond from Andrew Howroyd, Feb 21 2022

A377640 Number edge cuts in the n-cocktail party graph.

Original entry on oeis.org

0, 11, 1440, 2107526, 42988931520, 13507580252824616, 66106397544068778570240, 5070456260407959009146349277616, 6125036729796414112360988978198319006720, 116919909287476066493023898690514431465112835903616, 35352620147125599712871920668133699973012391980925543664619520
Offset: 1

Views

Author

Eric W. Weisstein, Nov 03 2024

Keywords

Links

Crossrefs

Cf. A297029.

Programs

  • PARI
    B(m,n) = {16^binomial(m,2) * 2^binomial(n,2) * 4^(m*n)}
S(m,n)={ my(M=matrix(m+1,n+1,i,j,B(i-1,j-1)), N=matrix(m+1,n+1));
  for(n=0, n, N[1,1+n] = M[1,1+n] - sum(j=1, n-1, binomial(n-1,j-1)*N[1,1+j]*M[1,1+n-j]));
  for(m=1, m, for(n=0, n, N[1+m, 1+n] = M[1+m,1+n] - sum(i=1, m, sum(j=0, n, binomial(m-1,i-1)*binomial(n,j)*N[1+i,1+j]*M[1+m-i,1+n-j]))));
  N
}
seq(n)=my(M=S(n,2*n)); vector(n, n, if(n==1, 0, 16^binomial(n,2) - sum(k=0, n, (-1)^k*binomial(n,k)*M[1+k, 1+2*(n-k)]))) \\ Andrew Howroyd, Jun 12 2025
  • PARI
    D(n,i)={16^binomial(i,2)*sum(j=0, 2*(n-i), 2^binomial(j,2)*4^(i*j)*x^j/j!, O(x^(2*(n-i)+1)))}
E(n)={my(u=vector(n+1,i,D(n,i-1)), v=vector(n+1)); v[1]=1+log(u[1]); for(m=1, n, v[1+m] = (u[1+m] - sum(i=1, m-1, binomial(m-1,i-1)*v[1+i]*u[1+m-i]))/u[1] ); v}
seq(n)={my(u=E(n)); vector(n, n, if(n==1, 0, 16^binomial(n,2) - sum(k=0, n, (-1)^k*binomial(n,k)*(2*n-2*k)!*polcoef(u[1+k], 2*(n-k)) )))} \\ Andrew Howroyd, Jun 12 2025

Extensions

a(5)-a(7) from Andrew Howroyd, May 27 2025
a(8) onwards from Andrew Howroyd, Jun 12 2025
Showing 1-2 of 2 results.

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

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