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

Author

Ralf Stephan, May 09 2007

Keywords

• nonn
• tabl

Comments

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

Examples

			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,
		

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.

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.

Programs

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 *)

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)))

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

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009