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

1,5

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.

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)

Example

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,...
...

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 *)

Prog

(PARI):
adu(M)=
{ /* Read by AntiDiagonals, Upwards */
    local(N=matsize(M)[1]);
    for (n=1, N, for(j=0, n-1, print1(M[n-j, j+1], ", ") ) );
}
T(n, k)=
{
    if ( (n==1) || (k==1), return(1) );
    return( sumdiv(n, d, d*T(d, k-1)) );
}
M=matrix(15, 15, n, k, T(n, k)) /* square array */
adu(M) /* sequence */

Crossrefs

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

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

Transposed array: A128119.

Sequence in context: A208344 A209172 A263950 * A345279 A376335 A342447

Adjacent sequences: A160867 A160868 A160869 * A160871 A160872 A160873

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Nov 19 2009

Status

approved