cp's OEIS Frontend

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

%I A132101 #54 Jul 15 2024 10:21:06
%S A132101 1,1,3,11,65,513,5363,68219,1016481,17243105,327431363,6874989963,
%T A132101 158118876449,3952936627361,106729080101235,3095142009014843,
%U A132101 95949394016339393,3166329948046914369,110821547820208233731,4100397266856761733515
%N A132101 a(n) = (A001147(n) + A047974(n))/2.
%C A132101 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 A132101 See the Burns et al. papers for another interpretation.
%C A132101 From _Ross Drewe_, Mar 16 2008: (Start)
%C A132101 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 A132101 a(n) = A000680: element order is significant and the labels are distinguishable;
%C A132101 b(n) = A001147: element order is significant but labels are not distinguishable, i.e., all label permutations of a given sequence are equivalent;
%C A132101 c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;
%C A132101 d(n) = A047974: reversal allowed, all label permutations are equivalent and equivalence class maps to itself under joint operation.
%C A132101 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 A132101 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 A132101 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 A132101 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 A132101 G. C. Greubel, <a href="/A132101/b132101.txt">Table of n, a(n) for n = 0..400</a>
%H A132101 Jonathan Burns, <a href="http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt">Assembly Graph Words - Single Transverse Component (Counts)</a>.
%H A132101 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 A132101 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 A132101 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 A132101 E.g.f.: (1/2)*( exp(x+x^2) + 1/sqrt(1-2*x) ). - _G. C. Greubel_, Jul 12 2024
%e A132101 a(2)=3 counts the arrangements [1122], [1212] and [1221]. - _R. J. Mathar_, Oct 18 2019
%p A132101 A132101 := proc(n)
%p A132101     (A001147(n)+A047974(n))/2 ;
%p A132101 end proc:
%p A132101 seq(A132101(n),n=0..30) ; # _R. J. Mathar_, Dec 20 2020
%t A132101 Table[((2n-1)!!+I^(-n)*HermiteH[n,I/2])/2,{n,0,30}] (* _Jonathan Burns_, Apr 05 2016 *)
%o A132101 (Magma)
%o A132101 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A132101 Coefficients(R!(Laplace( (Exp(x+x^2) + 1/Sqrt(1-2*x))/2 ))); // _G. C. Greubel_, Jul 12 2024
%o A132101 (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 A132101 Cf. A000680, A001147, A047974, A007769, A054499.
%Y A132101 Cf. A132100, A132101, A132102, A132103, A132104, A132105.
%K A132101 nonn,nice
%O A132101 0,3
%A A132101 _Keith F. Lynch_, Oct 31 2007
%E A132101 Entry revised by _N. J. A. Sloane_, Nov 04 2011