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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
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
Sequence in context: A111556 A300736 A226738 * A135905 A239296 A028498
KEYWORD
nonn
AUTHOR
Jonathan Burns, Apr 13 2016
STATUS
approved

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Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)