

A271212


a(n) = (2n1)*a(n1) + 2*(n2)*a(n2).


5



1, 2, 6, 34, 262, 2562, 30278, 419234, 6651846, 118950658, 2366492038, 51837444642, 1239591067526, 32130200470274, 897265598318022, 26856087563449762, 857662151219847238, 29108533617158451714, 1046243865439580921606, 39700713164247881457698, 1585992592561492290028038
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OFFSET

0,2


COMMENTS

a(n) is the number of reduced rearrangement maps on n blocks. A rearrangement map is a signed permutation, e.g., +2 1 3. If the permutation contains (i)(i+1) or (i+1)(i) for any i, then it is not reduced.
Number of permutations p of [2n] such that each element in p has exactly one neighbor whose value is smaller or larger by one. a(2) = 6: 1243, 2134, 2143, 3412, 3421, 4312.  Alois P. Heinz, May 24 2023


REFERENCES

J. Burns, Counting a Class of Signed Permutations and Rigid Vertex Graphs related to Patterns of DNA Rearrangement, Preprint.


LINKS



FORMULA

a(n) = (2n1)*a(n1) + 2*(n2)*a(n2); a(0)=1; a(1)=2;
a(n) = e^(1/2)*(2n+1)*Gamma(n,1/2)+(1)^n
a(n) = e^(1/2)*(2n+1)*2^(n1)*(n1)! + (1)^(n+1)*(2n^2 + 3n)^(1)* 2_F_2(1, n+1/2; n+1, n+5/2; 1/2)
a(n) = round( e^(1/2)*(2n+1)*2^(n1)*(n1)! )
a(n) ~ (Pi*2n/e)^(1/2) * (2n/e)^n
a(n) = Sum_{k = 0..n1} (1)^(n1+k)*2^(k+1)*(k+1)!*binomial(n1,k) for n >= 1.
2*exp(x)/(1  2*x)^2 = 2 + 6*x + 34*x^2/2! + 262*x^3/3! + 2562*x^4/4! + ... = Sum_{n >= 0} a(n+1)*x^n/n! is an e.g.f. for the sequence (a(n+1))n>=0.


EXAMPLE

For n=1 the a(1)=2 solutions are {+1,1}.
For n=2 the a(2)=6 solutions are {+12,1+2,12,+2+1,+21,2+1}. Note that {+1+2,21} are not reduced rearrangement maps.


MATHEMATICA

RecurrenceTable[{a[n]==(2n1)*a[n1]+2(n2)*a[n2], a[0]==1, a[1]==2}, a[n], {n, 0, 10}]
Table[Round[Exp[1/2]*(2n+1)*2^(n1)*(n1)!], {n, 10}]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



