A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

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

Offset

1,5

Comments

Differs from sum of divisors of m^(n-1) in 4th column!

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.

Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.

Formula

Dirichlet g.f. of n-th row: Product_{i=0..n-1} zeta(s-i).

If m is squarefree, T(n,m) = A000203(m^(n-1)). - Álvar Ibeas, Jan 17 2015

T(n, Product(p^e)) = Product(Gaussian_poly[e+n-1, e]_p). - Álvar Ibeas, Oct 31 2015

Example

Array starts:
1,1,1,1,1,1,1,1,1,
1,3,4,7,6,12,8,15,13,
1,7,13,35,31,91,57,155,130,
1,15,40,155,156,600,400,1395,1210,
1,31,121,651,781,3751,2801,11811,11011,
1,63,364,2667,3906,22932,19608,97155,99463,
1,127,1093,10795,19531,138811,137257,788035,896260,
1,255,3280,43435,97656,836400,960800,6347715,8069620,

Mathematica

T[n_, m_] := If[m == 1, 1, Product[{p, e} = pe; (p^(e+j)-1)/(p^j-1), {pe, FactorInteger[m]}, {j, 1, n-1}]];
Table[T[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 10 2018 *)

Prog

(PARI) T(n, m)=local(k, v); v=factor(m); k=matsize(v)[1]; prod(i=1, k, prod(j=1, n-1, (v[i, 1]^(v[i, 2]+j)-1)/(v[i, 1]^j-1)))

Crossrefs

Rows include A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

Columns include A000225, A003462, A006095, A003463, A160869, A023000, A006096, A006100, A046915.

Transposed array is A160870.

Sequence in context: A179745 A121300 A283595 * A158198 A158793 A112996

Adjacent sequences: A128116 A128117 A128118 * A128120 A128121 A128122

Keyword

nonn,tabl

Author

Ralf Stephan, May 09 2007

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

Status

approved