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 A271215 Number of loop-free assembly graphs with n rigid vertices. 2
 1, 0, 1, 4, 24, 184, 1911, 24252, 362199, 6162080, 117342912, 2469791336, 56919388745, 1425435420600, 38543562608825, 1119188034056244, 34733368101580440, 1147320305439301344, 40190943859500501151, 1488212241729974297796, 58080468361734193793551 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Number of chord diagrams (equivalent up to reflection) that do not contain any simple chords, e.g., 121332 contains the simple chord 33. REFERENCES J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint. LINKS Table of n, a(n) for n=0..20. Kristin DeSplinter, Satyan L. Devadoss, Jordan Readyhough, and Bryce Wimberly, Unfolding cubes: nets, packings, partitions, chords, arXiv:2007.13266 [math.CO], 2020. See Table 1 p. 15. FORMULA a(n) ~ (2n/e)^n / (e * sqrt(2)). a(n) = (|A000806(n)| + A271218(n)) / 2. a(n)/A132101(n) ~ 1/e. EXAMPLE For n=0 the a(0)=1 solution is { ∅ }. For n=1, a(1)=0 since the only assembly graph with one rigid vertex is the loop 11. For n=2, the a(2)=1 solution is { 1212 }. For n=3, the a(3)=4 solutions are { 121323, 123123, 123231, 123132 }. MATHEMATICA (Table[Sum[Binomial[n, i]*(2*n-i)!/2^(n-i)*(-1)^(i)/n!, {i, 0, n}], {n, 0, 20}]+RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5], a[0]==1, a[1]==0, a[2]==1, a[3]==3, a[4]==12}, a[n], {n, 0, 20}])/2 PROG (PARI) f(n) = sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) ); \\ A000806 lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4]; ); vector(nn-1, n, (va[n] + abs(f(n-1)))/2); } \\ Michel Marcus, Jul 28 2020 CROSSREFS Cf. A000806, A132101, A271218. Sequence in context: A111556 A300736 A226738 * A135905 A239296 A028498 Adjacent sequences: A271212 A271213 A271214 * A271216 A271217 A271218 KEYWORD nonn AUTHOR Jonathan Burns, Apr 13 2016 STATUS approved

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