A006184 Number of cycles in the complement of a path. (Formerly M3100)

0, 0, 0, 0, 0, 3, 23, 153, 1077, 8490, 75234, 742710, 8084990, 96192405, 1241588865, 17277139383, 257810397243, 4106342523108, 69531388662932, 1247182219179900, 23622547999002444, 471129863595453495, 9868783491120925755, 216617163296681315685, 4971829898824570284305, 119096935551493905531438, 2972224576868227286710038, 77153543251103295197353938

Offset

0,6

Comments

Number of cycles in K_n - P_n. - Sean A. Irvine, Jan 17 2017

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Path Complement Graph

Formula

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

a(n) ~ (n-1)! / (2*exp(1)). - Vaclav Kotesovec, Apr 22 2018

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Cf. A302734.

Sequence in context: A079755 A197176 A264461 * A308677 A209011 A164536

Adjacent sequences: A006181 A006182 A006183 * A006185 A006186 A006187

Keyword

nonn

Author

N. J. A. Sloane

Extensions

a(0)-a(3) prepended, a(4) corrected, and more terms from Sean A. Irvine, Jan 17 2017

Status

approved