Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M3100 #28 Apr 23 2018 11:58:12
%S 0,0,0,0,0,3,23,153,1077,8490,75234,742710,8084990,96192405,
%T 1241588865,17277139383,257810397243,4106342523108,69531388662932,
%U 1247182219179900,23622547999002444,471129863595453495,9868783491120925755,216617163296681315685,4971829898824570284305,119096935551493905531438,2972224576868227286710038,77153543251103295197353938
%N Number of cycles in the complement of a path.
%C Number of cycles in K_n - P_n. - _Sean A. Irvine_, Jan 17 2017
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A006184/b006184.txt">Table of n, a(n) for n = 0..200</a>
%H F. C. Holroyd and W. J. G. Wingate, <a href="http://dx.doi.org/10.1016/S0012-365X(85)80003-0">Cycles in the complement of a tree or other graph</a>, Discrete Math., 55 (1985), 267-282.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>
%F a(n) = (1/2)*Sum_{k=3..n} Sum_{i=1..k} Sum_{j=0..k-i} (-1)^(k-i)*(i-1)!*2^j*binomial(n+i-k, i)*binomial(i, j)*binomial(k-i-1, k-i-j). - _Andrew Howroyd_, Apr 21 2018
%F a(n) ~ (n-1)! / (2*exp(1)). - _Vaclav Kotesovec_, Apr 22 2018
%t Array[(1/2)Sum[Sum[Sum[(-1)^(k - i) (i - 1)!*2^j*Binomial[# + i - k, i] Binomial[i, j] Binomial[k - i - 1, k - i - j], {j, 0, k - i}], {i, k}], {k, 3, #}] &, 28, 0] (* _Michael De Vlieger_, Apr 21 2018 *)
%t Table[Sum[(-1)^(k - i) Gamma[i] 2^j Binomial[n + i - k, i] Binomial[i, j] Binomial[k - i - 1, k - i - j], {k, 3, n}, {i, k}, {j, 0, k - i}]/2, {n, 20}] (* _Eric W. Weisstein_, Apr 23 2018 *)
%o (PARI) a(n)={sum(k=3, n, sum(i=1, k, sum(j=0, min(i,k-i), (-1)^(k-i)*(i-1)!*2^j*binomial(n+i-k, i)*binomial(i, j)*binomial(k-i-1, k-i-j))))/2} \\ _Andrew Howroyd_, Apr 21 2018
%Y Cf. A302734.
%K nonn
%O 0,6
%A _N. J. A. Sloane_
%E a(0)-a(3) prepended, a(4) corrected, and more terms from _Sean A. Irvine_, Jan 17 2017