A132101 a(n) = (A001147(n) + A047974(n))/2.

1, 1, 3, 11, 65, 513, 5363, 68219, 1016481, 17243105, 327431363, 6874989963, 158118876449, 3952936627361, 106729080101235, 3095142009014843, 95949394016339393, 3166329948046914369, 110821547820208233731, 4100397266856761733515

Offset: 0

Author

Keith F. Lynch, Oct 31 2007

Keywords

• nonn
• nice

Comments

Also, number of distinct Tsuro tiles which are digonal in shape and have n points per side. Turning over is not allowed. See A132100 for definition and comments.

See the Burns et al. papers for another interpretation.

From Ross Drewe, Mar 16 2008: (Start)

This is also the number of arrangements of n pairs which are equivalent under the joint operation of sequence reversal and permutations of labels. Assume that the elements of n distinct pairs are labeled to show the pair of origin, e.g., [1 1], [2 2]. The number of distinguishable ways of arranging these elements falls as the conditions are made more general:

a(n) = A000680: element order is significant and the labels are distinguishable;

b(n) = A001147: element order is significant but labels are not distinguishable, i.e., all label permutations of a given sequence are equivalent;

c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;

d(n) = A047974: reversal allowed, all label permutations are equivalent and equivalence class maps to itself under joint operation.

Those classes that do not map to themselves form reciprocal pairs of classes under the joint operation and their number is r(n). Then c = b - r/2 = b - (b - d)/2 = (b+d)/2. A formula for r(n) is not available, but formulas are available for b(n) = A001147 and d(n) = A047974, allowing an explicit formula for this sequence.

c(n) is useful in extracting structure information without regard to pair ordering (see example). c(n) terms also appear in formulas related to binary operators, e.g., the number of binary operators in a k-valued logic that are invertible in 1 operation.

a(n) = (b(n) + c(n))/2, where b(n) = (2n)!/(2^n * n!) = A001147(n), c(n) = Sum_{k=0..floor(n/2)} n!/((n-2*k)! * k!) = A047974(n).

For 3 pairs, the arrangement A = [112323] is the same as B = [212133] under the permutation of the labels [123] -> [312] plus reversal of the elements, or vice versa. The unique structure common to A and B is {1 intact pair + 2 interleaved pairs}, where the order is not significant (contrast A001147). (End)

Examples

			a(2)=3 counts the arrangements [1122], [1212] and [1221]. - _R. J. Mathar_, Oct 18 2019
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
• Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394.
• R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section IV.

Crossrefs

Cf. A000680, A001147, A047974, A007769, A054499.

Cf. A132100, A132101, A132102, A132103, A132104, A132105.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( (Exp(x+x^2) + 1/Sqrt(1-2*x))/2 ))); // G. C. Greubel, Jul 12 2024

Maple

A132101 := proc(n)
    (A001147(n)+A047974(n))/2 ;
end proc:
seq(A132101(n),n=0..30) ; # R. J. Mathar, Dec 20 2020

Mathematica

Table[((2n-1)!!+I^(-n)*HermiteH[n,I/2])/2,{n,0,30}] (* Jonathan Burns, Apr 05 2016 *)

SageMath

[(factorial(n)*binomial(2*n,n) + (-2*i)^n*hermite(n,i/2))/2^(n+1) for n in range(31)] # G. C. Greubel, Jul 12 2024

Formula

D-finite with recurrence -(n-3)*a(n) +2*(n^2-3*n+1)*a(n-1) -(n-1)*a(n-2) -2*(2*n-5)*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Dec 20 2020

E.g.f.: (1/2)*( exp(x+x^2) + 1/sqrt(1-2*x) ). - G. C. Greubel, Jul 12 2024

Extensions

Entry revised by N. J. A. Sloane, Nov 04 2011