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A271218 Number of symmetric linked diagrams with n links and no simple link. 2
1, 0, 1, 3, 12, 39, 167, 660, 3083, 13961, 70728, 355457, 1936449, 10587960, 61539129, 361182139, 2224641540, 13880534119, 90090083047, 593246514588, 4038095508691 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Number of symmetric chord diagrams (where reflection is equivalent) with n chords and no simple chords.

Number of symmetric assembly words that do not contain the subword aa.

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.

FORMULA

a(n) = 2*a(n-1) + (2n-3)*a(n-2) - (2n-5)*a(n-3) + 2*a(n-4) - a(n-5)

a(n) = a(n-1) + 2*(n-1)*a(n-2) + a(n-3) + a(n-4) + 2*sum( k=0..n-4, a(k) )

a(n) ~ 2^(-1/2) * e^(-5/8) * (2n/e)^(n/2) * e^( sqrt(n/2) )  (conjectured)

a(n)/a(n-1) ~ sqrt(2n)  (conjectured)

EXAMPLE

For n=0 the a(0)=1 solution is { ∅ }

For n=1 there are no solutions since the link in a diagram with one link, 11, is simple

For n=2 the a(2)=1 solution is { 1212 }

For n=3 the a(3)=3 solutions are { 123123, 121323, 123231 }

For n=4 the a(4)=12 solutions are { 12123434, 12132434, 12324341, 12314234, 12341234, 12342341, 12314324, 12341324, 12343412, 12343421, 12324143, 12342143 }

MATHEMATICA

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, 20}]

CROSSREFS

A271218 / A047974 ~ 1/sqrt(e)  (conjectured).

Cf. A271215.

Sequence in context: A110153 A183366 A122994 * A062311 A303348 A237036

Adjacent sequences:  A271215 A271216 A271217 * A271219 A271220 A271221

KEYWORD

nonn,easy

AUTHOR

Jonathan Burns, Apr 13 2016

STATUS

approved

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Last modified January 17 15:12 EST 2020. Contains 330958 sequences. (Running on oeis4.)