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 A006184 Number of cycles in the complement of a path. (Formerly M3100) 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 * A209011 A164536 A037789 Adjacent sequences:  A006181 A006182 A006183 * A006185 A006186 A006187 KEYWORD nonn AUTHOR EXTENSIONS a(0)-a(3) prepended, a(4) corrected, and more terms from Sean A. Irvine, Jan 17 2017 STATUS approved

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Last modified July 19 14:33 EDT 2018. Contains 312776 sequences. (Running on oeis4.)