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A132101 a(n) = (A001147(n) + A047974(n))/2. 12
1, 1, 3, 11, 65, 513, 5363, 68219, 1016481, 17243105, 327431363, 6874989963, 158118876449, 3952936627361, 106729080101235, 3095142009014843, 95949394016339393, 3166329948046914369, 110821547820208233731, 4100397266856761733515 (list; graph; refs; listen; history; text; internal format)



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, eg, 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=0,1,...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)


Table of n, a(n) for n=0..19.

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, May 23, 2011.

R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section IV.


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


Table[((2n-1)!!+I^(-n)*HermiteH[n, I/2])/2, {n, 0, 10}] (* Jonathan Burns, Apr 05 2016 *)


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

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

Sequence in context: A030226 A309174 A233099 * A280775 A303341 A077428

Adjacent sequences:  A132098 A132099 A132100 * A132102 A132103 A132104




Keith F. Lynch, Oct 31 2007


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



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Last modified November 19 16:06 EST 2019. Contains 329320 sequences. (Running on oeis4.)