A378431
Number of cyclic edge cuts in the n-barbell graph.
Original entry on oeis.org
1, 727, 580369, 943372583, 4393791394153, 73031427044945215, 4761040298703872985897, 1242974526675429334954640663, 1300665117895338817158343055376913, 5449895182361696580039591706226169477735, 91388720915969010273161845143264574586208701497
Offset: 3
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scs(p,q=p) = {serconvol(serlaplace(p),serlaplace(q))}
seq(n)={my(g=sum(k=0,n,2^binomial(k,2)*x^k/k!, O(x*x^n)), t=sum(k=1,n,k^(k-2)*x^k/k!, O(x*x^n))); Vec( 4*scs(exp(t), g - (1+log(g)-t)*exp(t)) + 2*scs(g-exp(t)) - scs(intformal(exp(t)*deriv(log(g)-t))) )} \\ Andrew Howroyd, May 27 2025
A089459
Shifts 2 places left under the hyperbinomial transform.
Original entry on oeis.org
1, 1, 1, 2, 6, 30, 221, 2201, 27769, 424757, 7639190, 157969167, 3692733181, 96293669499, 2771232779481, 87247362424120, 2982901522141490, 110057420712635526, 4358745145279372293, 184433423940319376323
Offset: 0
A275488
Number of labeled forests of (free) trees such that exactly one tree is a path.
Original entry on oeis.org
1, 1, 3, 12, 80, 810, 10857, 174944, 3243060, 67859010, 1586109305, 41085509652, 1170954002946, 36469499267474, 1233416773419495, 45037748851872240, 1766375778253548392, 74067278799492363330, 3306928891056821667045, 156635771633727023132300
Offset: 1
a(1),a(2),a(3),a(4) are just a single path through an empty forest. a(5)=80 counts the 60 labelings of a path on 5 nodes and the 20 labelings of a path on 1 node and a star on 4 nodes.
- J. Harris, J. Hirst, M. Mossinghoff, Combinatorics and Graph Theory, Springer, 2010, page 34.
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nn = 20; b[z_] := 1/((1 - z) 2) - 1/2 + z/2;
t[z_] := z + Sum[n^(n - 2) z^n/n!, {n, 2, nn}];
Drop[Range[0, nn]! CoefficientList[Series[b[z] Exp[t[z] - b[z]], {z, 0, nn}], z], 1]
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