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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Greyson C. Wesley

Greyson C. Wesley's wiki page.

Greyson C. Wesley has authored 3 sequences.

A355511 a(n) is the number of monic polynomials of degree n over GF(11) without linear factors.

Original entry on oeis.org

0, 0, 55, 440, 5170, 56408, 620950, 6830120, 75131485, 826446280, 9090909091, 100000000000, 1100000000000, 12100000000000, 133100000000000, 1464100000000000, 16105100000000000, 177156100000000000, 1948717100000000000, 21435888100000000000, 235794769100000000000
Offset: 0

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Author

Greyson C. Wesley, Jul 04 2022

Keywords

Examples

			a(0) = 0 since there are no irreducible constant polynomials (as GF(11) is a field).
a(1) = 0 since all polynomials of degree 1 have linear factors.
a(2), the number of quadratic polynomials without linear factors, then coincides with the number of irreducible quadratics in GF(11), which is known to be M(11,2), where M(a,d) is the necklace polynomial, so a(2) = 55.

Crossrefs

Cf. A355510.

Programs

  • Mathematica
    necklacePolynomial[q_, n_] :=
  necklacePolynomial[q, n] = (1/n)*
    DivisorSum[n, MoebiusMu[n/#1]*q^#1 & ];
numIrreds[q_, n_] := If[n != 0, necklacePolynomial[q, n], 0];
restrictedPolynomialsOGF[q_, n_, d_] :=
  Product[(1 - z^If[ArrayDepth[d[[l]]] == 0, d[[l]], d[[l]][[1]]])^
     If[ArrayDepth[d[[l]]] == 0, numIrreds[q, d[[l]]],
             d[[l]][[2]]], {l, 1, Length[d]}]/(1 - q*z);
numRestrictedPolys[q_, n_, d_] :=
  SeriesCoefficient[restrictedPolynomialsOGF[q, n, d], {z, 0, n}];
q = 11;
TableForm[{#, numRestrictedPolys[q, #, {1}]} & /@ (Range[20]),
TableHeadings -> {{Row[{"(q=", q, ")"}]}, {"n", "#rootless monics"}}]

Formula

O.g.f. (1 - z)^(11)/(1-11*z) - 1

A355510 a(n) is the number of monic polynomials of degree n over GF(7) without linear factors.

Original entry on oeis.org

0, 0, 21, 112, 819, 5712, 39991, 279936, 1959552, 13716864, 96018048, 672126336, 4704884352, 32934190464, 230539333248, 1613775332736, 11296427329152, 79074991304064, 553524939128448, 3874674573899136, 27122722017293952
Offset: 0

Views

Author

Greyson C. Wesley, Jul 04 2022

Keywords

Examples

			a(0) = 0 since all constant polynomials are units (as GF(7) is a field).
a(1) = 0 since all polynomials of degree 1 have linear factors.
a(2), the number of quadratic polynomials without linear factors, then coincides with the number of irreducible quadratics in GF(7), which is known to be M(7,2), where M(a,d) is the necklace polynomial, so a(2) = 21.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1-z)^7/(1-7 z)-1,{z,0,15}]//Normal,z] (* For all terms *)
(7^(#-7)) &/@ Range[7,15]*6^7 (* For n>=7 *)
Join[{0,0,21,112,819,5712,39991},NestList[7#&,279936,20]] (* Harvey P. Dale, Oct 29 2022 *)

Formula

O.g.f.: (1 - z)^7/(1 - 7*z) - 1.
For n >= 7, a(n) = 6^7 * 7^(n-7).

A347018 The number of unlabeled trees T on n vertices for which maximum multiplicity attained by any matrix whose graph is T implies the simplicity of its other eigenvalues.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 14, 24, 43, 74, 134, 238, 433, 778, 1416, 2564, 4676, 8498, 15507, 28246, 51568, 94049, 171734, 313417, 572377, 1044986, 1908527, 3485092, 6365294, 11624741, 21232255, 38778177, 70828006, 129363233, 236282260, 431563697, 788254745, 1439742242
Offset: 1

Views

Author

Greyson C. Wesley, Aug 10 2021

Keywords

Comments

The trees counted by a(n) are called NIM trees on n vertices (No Intermediate Multiplicities). Equivalently, a(n) counts the trees on n vertices such that every vertex v of degree >= 3 satisfies two conditions:
(i) at most two branches at v have more than one vertex, and
(ii) at v has at least three adjacent vertices with degrees <= 3.
This is from the monograph "Eigenvalues, multiplicities and graphs" (see references).
Notice that they are restricted caterpillars: NIM trees are caterpillars on n vertices such that the following two conditions hold:
(i) Any degree sequence (...,a,b,c,...) for a>=4, b=4, c>=4 cannot be found in the degree sequence of the central path.
(ii) Any degree sequence (...,a,b,...) for a=3, b>=3, cannot be found in the degree sequence of the central path.

Examples

			By degree sequence of the central path:
a(1): (0)
a(2): (1,1)
a(3): (1,2,1)
a(4): (1,2,2,1),(1,3,1)
a(5): (1,2,2,2,1),(1,2,3,1),(1,4,1)
a(6): (1,2,2,2,2,1),(1,2,3,2,1),(1,2,2,3,1),(1,2,4,1),(1,5,1)
a(7): (1,2,2,2,2,2,1),(1,2,4,2,1),(1,2,2,4,1),(1,2,2,3,2,1),(1,2,2,2,3,1),
      (1,2,5,1),(1,3,2,3,1),(1,6,1)
a(8): (1,2,2,2,2,2,2,1),(1,3,2,2,2,2,1),(1,2,3,2,2,2,1),(1,2,2,3,2,2,1),
      (1,4,2,2,2,1),(1,2,4,2,2,1),(1,5,2,2,1),(1,2,5,2,1),
      (1,6,2,1),(1,3,2,3,2,1),(1,3,2,2,3,1),(1,7,1),
      (1,4,2,3,1), (1,4,4,1)

References

  • Charles R. Johnson and Carlos M. Saiago, Eigenvalues, multiplicities and graphs, Cambridge University Press, 2018.

Links

Programs

  • Mathematica
    A[z_,u_,r_]:=r z^4 (1+u z+(u^2 z^4)/(1-u z^4));
Nsk[z_,u_,r_]:=r z(1/2 (1/(1-1/z A[z,u,r])+(1+1/z A[z,u,r])/(1-1/z^2 A[z^2,u^2,r^2]))-1)
alpha[z_,u_,r_]:=r A[z,u,r]+(2Nsk[z^2,u^2,r^2])/(z r) (1+A[z,u,r]/z);
beta[z_,u_,r_]:=-1/(r z) (1+A[z,u,r]/z);
Nsksym[z_,u_,r_]:=Sum[Product[beta[z^2^i,u^2^i,r^2^i],{i,0,k-1}] alpha[z^2^k,u^2^k,r^2^k],{k,0,6}];
Nskasym[z_,u_,r_]:=Nsk[z,u,r]-Nsksym[z,u,r];
Nsksymodu[z_,u_,r_]:=(Nsksym[z,u,r]-Nsksym[z,-u,r])/2;
Nsksymodr[z_,u_,r_]:=(Nsksym[z,u,r]-Nsksym[z,u,-r])/2;
Nsksymevuevr[z_,u_,r_]:=(Nsksym[z,u,r]+Nsksym[z,-u,r]+Nsksym[z,u,-r]+Nsksym[z,-u,-r])/4;
Nsym[z_]:=Nsksymevuevr[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]]+(Sqrt[1-z^2]/(1-z))(Nsksymodr[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]])+(Sqrt[1-z^2]/(1-z))(Nsksymodu[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]]);
Nim[z_]:=(z/(1-z)+Nsym[z])+(Nskasym[z,1/(1-z),1/(1-z)]+1/2(Nsksymevuevr[z,1/(1-z),1/(1-z)]-Nsksymevuevr[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]])+1/2 (Nsksymodr[z,1/(1-z),1/(1-z)]-Sqrt[1-z^2]/(1-z) Nsksymodr[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]])+1/2 (Nsksymodu[z,1/(1-z),1/(1-z)]-Sqrt[1-z^2]/(1-z) Nsksymodu[z,1/Sqrt[1-z^2],1/Sqrt[1-z^2]]));
TableForm[Table[{i,Coefficient[Series[Nim[z]/.{u -> 1, r -> 1}, {z, 0, 50}], z^i]}, {i,1, 50}]]

Formula

Conjecture: For n>15, a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + 2*a(n-4) - a(n-5) - 2*a(n-6) + a(n-7) + 3*a(n-8) - 4*a(n-9) - a(n-10) + 2*a(n-11) - 2*a(n-12) + 2*a(n-13) + a(n-14) - a(n-15), with initial terms a(1) through a(15) as given.
If the above conjecture holds, then a(n) has o.g.f. (x - x^2 - 2*x^3 + 2*x^4 - x^5 - x^6 + 2*x^7 - 3*x^9 + 2*x^10 + x^11 - x^12 + 2*x^13 - x^15)/(1 - 2*x - x^2 + 3*x^3 - 2*x^4 + x^5 + 2*x^6 - x^7 - 3*x^8 + 4*x^9 + x^10 - 2*x^11 + 2*x^12 - 2*x^13 - x^14 + x^15).
If the above conjecture holds, then a(n) has an explicit Binet-like formula that can be obtained in terms of the roots of a fifth-degree polynomial.

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