A132100 -id:A132100 - cp's OEIS frontend

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

Keith F. Lynch, Oct 31 2007

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)

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

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.

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

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

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

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

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

[(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

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

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

A132105 Number of distinct Tsuro tiles which are n-gonal in shape and have 2 points per side.

Original entry on oeis.org

1, 1, 3, 7, 30, 137, 1065, 10307, 130040, 1927853, 32809979, 625303343, 13178378742, 304081128617, 7623562484349, 206343110670031, 5996839161108904, 186254714746749377, 6156752738537004317, 215810382975655205399, 7995774673152799224930

Offset: 0

Keith F. Lynch, Oct 31 2007

nonn

Turning over is allowed.

See A132100 for definition and comments.

Cf. A132100-A132104, A007769, A001147, A054499.

# B(n,m) gives the number of n-sided tiles with m points per side, allowing reflections (cf. comments and formula of A132100)
with(numtheory): a:=(p,r)->piecewise(p mod 2 = 1,p^(r/2)*doublefactorial(r-1), sum(p^j*binomial(r, 2*j)*doublefactorial(2*j - 1), j = 0 .. floor(r/2)));
B := (n,m)->piecewise(n*m mod 2=1,0,add(phi(p)*a(p,m*n/p),p in divisors(n))/(2*n)+
piecewise(m mod 2=0, a(2,m*n/2)*2, a(2,m*n/2)+a(2,m*n/2-1))/4);
A132105 := n -> B(n,2);[seq(A132105(n),n=1..20)]; # Laurent Tournier, Jul 09 2014

More terms from Laurent Tournier, Jul 09 2014

A132104 Number of distinct Tsuro tiles which are square and have Q points per side.

Original entry on oeis.org

1, 2, 30, 1447, 257107, 81898020, 39531524384, 26682303327353, 23987350539183237, 27705387002314059046, 39978873351170263411714, 70482753710219315731386411, 149071024096816130023228547735, 372528489217914304271725034290952, 1085920546070218942128273877774286532, 3651950796434146162433577686485443037885

Offset: 0

Keith F. Lynch, Oct 31 2007

nonn

Turning over is allowed.

See A132100 for definition and comments.

Cf. A132100-A132105, A007769, A001147, A054499.

# B(n,m) gives the number of n-sided tiles with m points per side, allowing reflections (cf. comments in A132100)
with(numtheory): a:=(p,r)->piecewise(p mod 2 = 1,p^(r/2)*doublefactorial(r-1), sum(p^j*binomial(r, 2*j)*doublefactorial(2*j - 1), j = 0 .. floor(r/2)));
B := (n,m)->piecewise(n*m mod 2=1,0,add(phi(p)*a(p,m*n/p),p in divisors(n))/(2*n) + piecewise(m mod 2=0, a(2,m*n/2)*2, a(2,m*n/2)+a(2,m*n/2-1))/4);
A132104 := m -> B(4,m);[seq(A132104(m),m=1..15)]; # Laurent Tournier, Jul 09 2014

More terms from Laurent Tournier, Jul 09 2014

A132102 Number of distinct Tsuro tiles which are n-gonal in shape and have 2 points per side.

Original entry on oeis.org

1, 1, 3, 7, 35, 193, 1799, 19311, 254143, 3828921, 65486307, 1249937335, 26353147811, 608142583137, 15247011443103, 412685556939751, 11993674252049647, 372509404162520641, 12313505313357313047, 431620764875678503143, 15991549339008732109899

Offset: 0

Keith F. Lynch, Oct 31 2007

nonn

Turning over is not allowed.
See A132100 for definition and comments.
Even and odd terms can be computed with the help of Burnside Lemma and recursive sequences. - Lionel RAVEL, Sep 18 2013

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001147, A007769, A054499, A132100, A132101, A132103, A132104, A132105.

with(numtheory): a:=(p,q)->piecewise(p mod 2 = 1, p^q*doublefactorial(2*q - 1), sum(p^j*binomial(2*q, 2*j)*doublefactorial(2*j - 1), j = 0 .. q));
A132102 := n->add(phi(p)*a(p,n/p),p in divisors(n))/n;
[seq(A132102(n),n=1..20)]; # Laurent Tournier, Jul 09 2014

a(n)={if(n<1, n==0, sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(j=0, m, (d%2==0 || m-j==0) * binomial(2*m, 2*j) * d^j * (2*j)! / (j!*2^j) ))/n)} \\ Andrew Howroyd, Jan 26 2020

a(n) = (1/n)*Sum_{d|n} phi(d)*alpha(d, n/d), where phi = Euler's totient function,
alpha(p,q) = Sum_{j=0..q} p^j * binomial(2q, 2j) * (2j-1)!! if p even,
= p^q * (2q-1)!! if p odd. (cf. also A132100) - Laurent Tournier, Jul 09 2014

More terms from Lionel RAVEL, Sep 18 2013

a(9) and a(10) corrected, and addition of more terms using formula given above by Laurent Tournier, Jul 09 2014

cp's OEIS Frontend

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

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A132105 Number of distinct Tsuro tiles which are n-gonal in shape and have 2 points per side.

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A132104 Number of distinct Tsuro tiles which are square and have Q points per side.

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A132102 Number of distinct Tsuro tiles which are n-gonal in shape and have 2 points per side.

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A132103 Number of distinct Tsuro tiles which are digonal in shape and have Q points per side.

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