A378324 Number of cyclic edge cuts in the complete graph K_n.

0, 0, 0, 0, 0, 10, 840, 57428, 4323760, 428530774, 66698370662, 19304350714396, 11435576322977378, 13998454986272457974, 34730539006860778387488, 172307954877667746584363616, 1699711619922134215461075979752, 33269167602899548362529088074829390

Offset: 1

Author

Eric W. Weisstein, Nov 23 2024

Keywords

• nonn

Links

• Eric W. Weisstein, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Complete Graph.
• Eric Weisstein's World of Mathematics, Cyclic Edge Cut.

Crossrefs

Cf. A000272, A001187, A006125, A001858.

Programs

Mathematica

With[{n = 20},
 B[x_] := 1 + Log[Sum[2^Binomial[k, 2] x^k/k!, {k, 0, n}]];
 T[x_] := 1 - LambertW[-x] - LambertW[-x]^2/2;
 Rest@CoefficientList[Series[(Exp[B[x] - T[x]] - (B[x] - T[x]) - 1) Exp[T[x] - 1], {x, 0, n}], x] Range[n]!
]

PARI

seq(n)={my(t=sum(k=1, n, k^(k-2)*x^k/k!, O(x*x^n)), c=log(sum(k=0, n, 2^binomial(k,2)*x^k/k!, O(x*x^n)))-t); Vec(serlaplace((exp(c)-1-c)*exp(t)), -n)} \\ Andrew Howroyd, Nov 26 2024

Formula

E.g.f.: (exp(B(x)-T(x))-(B(x)-T(x))-1)*exp(T(x)-1) where B(x) is the e.g.f. of A001187 and T(x) is the e.g.f. of A000272. - Andrew Howroyd, Nov 26 2024

a(n) = A006125(n) - A001858(n) - Sum_{k=1..n} binomial(n,k)*(A001187(k)-A000272(k))*A001858(n-k). - Andrew Howroyd, May 27 2025

Extensions

a(8) onwards from Andrew Howroyd, Nov 26 2024