A039623 -id:A039623 - cp's OEIS frontend

A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

1, 4, 7, 10, 15, 18, 21, 24, 29, 36, 39, 42, 47, 50, 53, 56, 65, 68, 75, 78, 83, 86, 89, 92, 97, 108, 111, 118, 123, 126, 129, 132, 141, 144, 147, 150, 163, 166, 169, 172, 177, 180, 183, 186, 191, 198, 201, 204, 213, 228, 239, 242, 247, 250, 257, 260, 265, 268

Offset: 0

Vladeta Jovovic, Sep 27 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000010, A000188, A064276, A059306, A062801, A059976, A039623.

Cf. A001620, A306016.

f[p_, e_] := p^Floor[e/2]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 100}, 1 + Range[0, max] + 2 * Accumulate[Array[a, max + 1, 0]]] (* Amiram Eldar, Nov 07 2024 *)

a(n) = n + 1 + 2*sum(k=1, n, sumdiv(k, d, issquare(d)*eulerphi(sqrtint(d)))) \\ Michel Marcus, Jun 17 2013

a(n) = n + 1 + 2*Sum_{k=1..n} Sum_{d^2|k} phi(d), where phi = Euler totient function A000010.

a(n) ~ (n/zeta(2)) * (log(n) + 3*gamma - 1 + zeta(2) - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 07 2024

A064363 Number of 2 X 2 regular integer matrices with elements from {0,...,n} up to row and column permutation.

Original entry on oeis.org

0, 2, 14, 51, 133, 289, 547, 954, 1546, 2380, 3508, 5005, 6915, 9347, 12353, 16028, 20468, 25790, 32054, 39427, 47965, 57833, 69155, 82082, 96682, 113192, 131720, 152429, 175467, 201075, 229305, 260492, 294700, 332182, 373138, 417751, 466201

Offset: 0

Vladeta Jovovic, Sep 25 2001

			There are 2 binary regular matrices up to row and column permutation:
[1 0] [1 1]
[0 1] [1 0].

Cf. A064276, A059306, A062801, A059976, A039623.

A059306[0] = 1; A059306[n_] := Table[{w, x, y, z} /. {ToRules[ Reduce[0 <= x <= n && 0 <= y <= n && 0 <= z <= n && w*z - x*y == 0, {x, y, z}, Integers]]}, {w, 0, n}] // Flatten[#, 1] & // Length; a[n_] := ((n + 1)*(n^3 + 3*n^2 + 4*n + 1) - A059306[n])/4; Table[Print[an = a[n]]; an, {n, 0, 36}] (* Jean-François Alcover, Nov 26 2013 *)

a(n) = ((n+1)*(n^3+3*n^2+4*n+1)-A059306(n))/4.

A367505 Triangle read by rows: row n gives the h-vector of the n-th halohedron.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 27, 13, 1, 1, 21, 76, 76, 21, 1, 1, 31, 175, 300, 175, 31, 1, 1, 43, 351, 925, 925, 351, 43, 1, 1, 57, 637, 2401, 3675, 2401, 637, 57, 1, 1, 73, 1072, 5488, 11956, 11956, 5488, 1072, 73, 1, 1, 91, 1701, 11376, 33516, 47628, 33516, 11376, 1701, 91, 1

Offset: 0

F. Chapoton, Nov 21 2023

Theorem 6.1.11 in Almeter's thesis gives the f-vector generating series. Then replacing x with x-1 gives the h-vector generating series.

			As a table:
  (1),
  (1,  1),
  (1,  3,  1),
  (1,  7,  7,  1),
  (1, 13, 27, 13,  1),
  (1, 21, 76, 76, 21,  1),
  ...

Jordan Grady Almeter, P-graph associahedra and hypercube graph associahedra, arXiv:2211.02113 [math.CO], 2022; Ph.D. thesis, North Carolina State University, 2022.
Forcey's Hedra Zoo, Halohedron.

Row sums are A051960(n-1) for n>=1.
Alternating sums form an aerated version of A110556.
Columns k=0-2 give A000012, A002061, A039623(n-1) for n>=2.

T[0,0]:=1;T[n_,k_]:= Binomial[n-1,n-k]*Binomial[n,n-k]+Binomial[n-1,n-k-1]^2;Flatten[Table[T[n,k],{n,0,10},{k,0,n}]] (* Detlef Meya, Nov 23 2023 *)

x = polygen(QQ, 'x')
t = x.parent()[['t']].0
F = (1 + (1+x) * t) / (2 * sqrt(1 - 2 * (x+1) * t + (x-1)**2 * t**2)) + 1/2
for poly in F.list(): print(poly.list())

G.f.: (1 + (1+x)*t)/(2*sqrt(1 - 2*(x+1)*t + (x-1)^2*t^2)) + 1/2.

T(0,0) = 1; T(n,k) = binomial(n-1,n-k)*binomial(n,n-k)+binomial(n-1,n-k-1)^2. - Detlef Meya, Nov 23 2023

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A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

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A064363 Number of 2 X 2 regular integer matrices with elements from {0,...,n} up to row and column permutation.

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A367505 Triangle read by rows: row n gives the h-vector of the n-th halohedron.

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