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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 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

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