A002995 -id:A002995 - cp's OEIS frontend

A003243 Number of partially achiral trees with n nodes.

1, 1, 1, 2, 3, 6, 9, 19, 30, 61, 99, 198, 333, 650, 1115, 2143, 3743, 7101, 12553, 23605, 42115, 78670, 141284, 262679, 474083, 878386, 1591038, 2940512, 5340712, 9852201, 17930619, 33031498, 60209609, 110801271, 202208576, 371820314

Offset: 1

N. J. A. Sloane

The g.f. (1-z**2-2*z**3-8*z**4+7*z**5+4*z**6)/(1-z-z**2-2*z**3-6*z**4+14*z**5) was conjectured by Simon Plouffe in his 1992 dissertation, but this is incorrect.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..3770 (terms 1..73 from Herman Jamke)
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to trees

t(n)=local(A=x); if(n<1, 0, for(k=1, n-1, A/=(1-x^k+x*O(x^n))^polcoeff(A, k)); polcoeff(A, n)) {n=100;Ty2=sum(i=0,n,t(i)*y^(2*i)); p=subst(y*Ty2/(y-Ty2),y,y+y*O(y^n));p=Pol(p,y);a=subst(Ty2*(y+p+(p^2-subst(p,y,y^2))/(2*y))/y^2-(p^2+subst(p,y,y^2))/(2*y^2)+Ty2,y,x+x*O(x^n)); for(i=0,n-2,print1(polcoeff(a,i)","))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008

a(n) ~ c * d^n, where d = 1.8332964415228533737988849634129366404833316666328290543862325494628120733... is the root of the equation Sum_{k>=1} A000081(k) / d^(2*k-1) = 1 and c = 0.123308773712306885475561730669251048497115967922743533462465528423705228... - Vaclav Kotesovec, Dec 13 2020

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008

A003244 Number of unrooted achiral trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 16, 23, 35, 51, 72, 97, 136, 186, 230, 321, 401, 526, 647, 844, 1000, 1331, 1539, 1960, 2299, 2943, 3307, 4237, 4779, 5961, 6744, 8372, 9239, 11605, 12694, 15549, 17264, 21086, 22784, 27976, 30357, 36598, 39843, 47821

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
Index entries for sequences related to trees

In terms of generating functions: A003244(x) = A003241(x)-(P^2(x)-P(x^2))/(2*x^2) with P(x)=x*A003238(x). [Harary & Robinson eq 45]. - R. J. Mathar, Sep 28 2011

Extended by R. J. Mathar, Sep 28 2011

A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 12, 8, 2, 10, 48, 64, 25, 3, 26, 196, 412, 314, 78, 6, 80, 798, 2458, 2976, 1478, 270, 14, 246, 3248, 13452, 23588, 18844, 6748, 926, 34, 810, 13184, 70330, 166050, 192096, 110714, 30168, 3305, 95, 2704, 53416, 353716, 1074472, 1676668, 1397484, 613884, 132734, 11868, 280, 9252

Offset: 1

Andrew Howroyd, Jan 21 2025

The number of edges is n + k - 2.

			Array begins:
==============================================================
n\k |  1    2      3      4       5       6       7      8 ...
----+---------------------------------------------------------
  1 |  1    1      2      4      10      26      80    246 ...
  2 |  1    3     12     48     196     798    3248  13184 ...
  3 |  1    8     64    412    2458   13452   70330 353716 ...
  4 |  2   25    314   2976   23588  166050 1074472 ...
  5 |  3   78   1478  18844  192096 1676668 ...
  6 |  6  270   6748 110714 1397484 ...
  7 | 14  926  30168 613884 ...
  8 | 34 3305 132734 ...
   ...

Columns 1..2 are A002995, A060404.
Rows 1..2 are A003239(n-1), A103943.
Antidiagonal sums are A103937.
Cf. A269920 (rooted), A379430 (sensed with no root).

A039791 Sequence arising in search for Legendre sequences.

Original entry on oeis.org

1, 1, 2, 4, 6, 14, 66, 95, 280, 1464, 2694, 10452, 41410, 95640, 323396, 1770963, 5405026, 13269146, 73663402, 164107650, 582538732, 3811895344, 7457847082, 30712068524, 151938788640, 353218528324, 1738341231644, 7326366290632, 17280039555348, 63583110959728

Offset: 1

N. J. A. Sloane

nonn

Number of bit strings of length L = 2n+1 and Hamming weight n (or n+1, as generated by Fletcher et al.) up to chord equivalence (i.e., up to color and general linear permutation x -> Ax+b mod L for A on Z/LZ* and b on Z/LZ--essentially a multiplicative necklace of phi(L) additive necklaces of L black and white beads where L is odd and the colors are as balanced as possible). The same strings are counted up to bracelet equivalence (x -> +-x+b mod L) at A007123, up to necklace equivalence (x -> x+b mod L) at A000108, and in full (x -> x) at A001700. - Travis Scott, Nov 24 2022

			From _Travis Scott_, Nov 24 2022: (Start)
If we decompose by weight the classes of period 2n+1 counted at A002729, a(n) appears as the twin towers of that triangle.
                              a(n)
                             |   |
                            (1) (1)
                         1   1   1   1
                     1   1   1   1   1   1
                 1   1   1   2   2   1   1   1
             1   1   2   3   4   4   3   2   1   1
         1   1   1   2   4   6   6   4   2   1   1  1
      1  1   1   3   7  10  14  14  10   7   3   1  1  1
   1  1  3   7  18  34  54  66  66  54  34  18   7  3  1  1
1  1  1  3  11  25  49  75  95  95  75  49  25  11  3  1  1  1. (End)

Roderick J. Fletcher, Marc Gysin, and Jennifer Seberry, Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices, Australasian J. Combin. 23 (2001), 75-86.

Coincides with A002995 offset by -1 at the A005097-th terms.

Cf. A000108, A007123, A001700.

Module[{a,b,g,L,m,x,z,Z},Table[L=2n+1;Z=Sum[Sum[Product[g=L/GCD[L,(k-1)i+j];Subscript[x,#]^(1/#)&@If[k==1,g,m=MultiplicativeOrder[k,g];g/GCD[g,(k^m-1)/(k-1)]m],{i,L}]L/GCD[L,k-1],{j,GCD[L,k-1]}],{k,Select[Range@L,CoprimeQ[#,L]&]}]/L/EulerPhi@L/.Subscript[x,z_]->a^z+b^z;CoefficientList[Z,{a,b}][[n+1,n+2]],{n,30}]] (* Travis Scott, Nov 24 2022 *)

a(n) ~ C(2n+1, n)/(2n+1)/phi(2n+1)

Empirical: a(n) == 1 (mod 2) for 2n+1 of the form 2^k+1 but not of the form p^2, else == 0.

More terms from Travis Scott, Nov 24 2022

cp's OEIS Frontend

A003243 Number of partially achiral trees with n nodes.

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A003244 Number of unrooted achiral trees with n nodes.

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A380240 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces including one distinguished outside face, n >= 1, k >= 1.

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A039791 Sequence arising in search for Legendre sequences.

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