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

Author

N. J. A. Sloane, Nov 19 2009

Keywords

• nonn
• tabl
• easy

Examples

			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,...
  ...
		

References

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

Links

• 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.

Crossrefs

Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.

Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.

Transposed array: A128119.

Programs

Mathematica

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 *)

PARI

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

Formula

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)