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A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Transposed array is A160870.
Sequence in context: A179745 A121300 A283595 * A158198 A158793 A112996
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, May 09 2007
EXTENSIONS
Edited by Charles R Greathouse IV, Oct 28 2009
STATUS
approved

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Last modified July 20 13:32 EDT 2024. Contains 374445 sequences. (Running on oeis4.)