A132101 - OEIS

%I #54 Jul 15 2024 10:21:06

%S 1,1,3,11,65,513,5363,68219,1016481,17243105,327431363,6874989963,

%T 158118876449,3952936627361,106729080101235,3095142009014843,

%U 95949394016339393,3166329948046914369,110821547820208233731,4100397266856761733515

%N a(n) = (A001147(n) + A047974(n))/2.

%C 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.

%C See the Burns et al. papers for another interpretation.

%C From _Ross Drewe_, Mar 16 2008: (Start)

%C 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:

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

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

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

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

%C 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 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.

%C 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).

%C 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)

%H G. C. Greubel, <a href="/A132101/b132101.txt">Table of n, a(n) for n = 0..400</a>

%H Jonathan Burns, <a href="http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt">Assembly Graph Words - Single Transverse Component (Counts)</a>.

%H Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, and Masahico Saito, <a href="https://doi.org/10.1016/j.dam.2013.01.003">Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination</a>, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394.

%H R. J. Mathar, <a href="http://vixra.org/abs/1901.0148">Feynman diagrams of the QED vacuum polarization</a>, vixra:1901.0148 (2019), Section IV.

%F 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

%F E.g.f.: (1/2)*( exp(x+x^2) + 1/sqrt(1-2*x) ). - _G. C. Greubel_, Jul 12 2024

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

%p A132101 := proc(n)

%p     (A001147(n)+A047974(n))/2 ;

%p end proc:

%p seq(A132101(n),n=0..30) ; # _R. J. Mathar_, Dec 20 2020

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

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o Coefficients(R!(Laplace( (Exp(x+x^2) + 1/Sqrt(1-2*x))/2 ))); // _G. C. Greubel_, Jul 12 2024

%o (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

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

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

%K nonn,nice

%O 0,3

%A _Keith F. Lynch_, Oct 31 2007

%E Entry revised by _N. J. A. Sloane_, Nov 04 2011