A128119 -id:A128119 - cp's OEIS frontend

A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 7, 13, 15, 1, 1, 6, 35, 40, 31, 1, 1, 12, 31, 155, 121, 63, 1, 1, 8, 91, 156, 651, 364, 127, 1, 1, 15, 57, 600, 781, 2667, 1093, 255, 1, 1, 13, 155, 400, 3751, 3906, 10795, 3280, 511, 1, 1, 18, 130, 1395, 2801, 22932, 19531, 43435, 9841, 1023, 1

Offset: 1

N. J. A. Sloane, Nov 19 2009

			Array begins:
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
  1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
  1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
  1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
  1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
  ...

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.
Yi Ming Zou, Gaussian binomials and the number of sublattices, Acta Cryst. A62 (2006) 409-410.

Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.
Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.
Transposed array: A128119.

T[, 1] = 1; T[1, ] = 1; T[n_, k_] := T[n, k] = DivisorSum[n, (n/#)^(k-1) *T[#, k-1]&]; Table[T[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2015 *)

T(n,k)={ if ( (n==1) || (k==1), 1, sumdiv(n,d, d*T(d, k-1)) ) }

T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
From Álvar Ibeas, Oct 31 2015: (Start)
T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1).
T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p). (End)

A300782 Number of symmetrically distinct sublattices (supercells, superlattices, HNFs) of the simple cubic lattice of index n.

Original entry on oeis.org

1, 3, 3, 9, 5, 13, 7, 24, 14, 23, 11, 49, 15, 33, 31, 66, 21, 70, 25, 89, 49, 61, 33, 162, 50, 81, 75, 137, 49, 177, 55, 193, 97, 123, 99, 296, 75, 147, 129, 312, 89, 291, 97, 269, 218, 203, 113, 534, 146, 302, 203, 357, 141, 451, 207, 508, 247, 307, 171, 789

Offset: 1

Andrey Zabolotskiy, Mar 12 2018

nonn

Andrey Zabolotskiy, Table of n, a(n) for n = 1..1000
Matt DeCross, Lattice Polytopes and Orbifolds, 2015.
Matt DeCross, Lattice Polytopes and Orbifolds in Quiver Gauge Theories, 2015. See slides 18-21.
Gus L. W. Hart and Rodney W. Forcade, Algorithm for generating derivative structures, Phys. Rev. B 77, 224115 (2008), DOI: 10.1103/PhysRevB.77.224115 [see Table IV].
Materials Simulation Group, Derivative structure enumeration library
Index entries for sequences related to sublattices
Index entries for sequences related to cubic lattice

Cf. A159842, A300783, A300784, A003051, A145393, A001001, A128119, A160870, A145396, A145398.

# see A159842 for the definition of dc, fin, per, u, N, N2
def a(n): # from DeCross's slides
    return (dc(u, N, N2)(n) + 6*dc(fin(1, -1, 0, 4), u, u, N)(n)
      + 3*dc(fin(1, 3), u, u, N)(n)
      + 8*dc(fin(1, 0, -1, 0, 0, 0, 0, 0, 3), u, u, per(0, 1, -1))(n)
      + 6*dc(fin(1, 1), u, u, per(0, 1, 0, -1))(n))//24
print([a(n) for n in range(1, 300)])
# Andrey Zabolotskiy, Sep 02 2019

Terms a(11) and beyond from Andrey Zabolotskiy, Sep 02 2019

cp's OEIS Frontend

A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

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