A351588 Number of minimal edge covers in the n-path complement graph.

0, 0, 0, 1, 7, 34, 174, 1079, 7055, 48796, 366180, 2928387, 24726556, 220572828, 2071469527, 20393131971, 209934610376, 2254860549906, 25210893460938, 292826210789807, 3527105947667676, 43985152403166462, 567048383126842506, 7546842245268945427, 103560659196050026908

Offset: 1

Author

Eric W. Weisstein, Feb 14 2022

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Minimal Edge Cover.
• Eric Weisstein's World of Mathematics, Path Complement Graph.

Crossrefs

Cf. A302719, A302749, A351587.

Programs

PARI

a(n)={sum(i=0, n\2, sum(j=0, (n-2*i)\3, my(r=n-2*i-3*j, g=exp(x + O(x*x^r))); (-1)^i*binomial(i+j,i)*binomial(n-i-2*j,i+j)*(r)!*polcoef((2*g-1)^i*exp(j*x -x - x^2/2 + x*g), r)))} \\ Andrew Howroyd, Jun 14 2025

Formula

a(n) = Sum_{i=0..floor(n/2)} Sum{j=0..floor((n-2*i)/3)} (-1)^i * binomial(i+j,i) * binomial(n-i-2*j,i+j) * (n-2*i-3*j)! * [x^(n-2*i-3*j)] ((2*exp(x)-1)^i * exp(x)^j * exp(-x - x^2/2 + x*exp(x))). - Andrew Howroyd, Jun 14 2025

Extensions

a(9)-a(12) from Andrew Howroyd, Feb 21 2022

a(13) onwards from Andrew Howroyd, Jun 14 2025