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%I #19 Dec 10 2018 08:52:56
%S 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,
%T 1,8,91,156,651,364,127,1,1,15,57,600,781,2667,1093,255,1,1,13,155,
%U 400,3751,3906,10795,3280,511,1,1,18,130,1395,2801,22932,19531,43435,9841,1023,1
%N Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).
%D Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]
%H Álvar Ibeas, <a href="/A160870/b160870.txt">First 100 antidiagonals, flattened</a>
%H Michael Baake, <a href="http://arxiv.org/abs/math/0605222">Solution of the coincidence problem in dimensions d≤4</a>, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
%H B. Gruber, <a href="https://doi.org/10.1107/S0108767397009781">Alternative formulas for the number of sublattices</a>, Acta Cryst. A53 (1997) 807-808.
%H J. H. Kwak and J. Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%H Yi Ming Zou, <a href="https://arxiv.org/abs/math/0610684">Gaussian binomials and the number of sublattices</a>, arXiv:math/0610684 [math.CO], 2006.
%H Yi Ming Zou, <a href="https://doi.org/10.1107/S010876730602455X">Gaussian binomials and the number of sublattices</a>, Acta Cryst. A62 (2006) 409-410.
%F T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
%F From _Álvar Ibeas_, Oct 31 2015: (Start)
%F T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1).
%F T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p).
%F (End)
%e Array begins:
%e 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,...
%e 1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
%e 1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
%e 1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
%e 1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
%e ...
%t 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 *)
%o (PARI):
%o adu(M)=
%o { /* Read by AntiDiagonals, Upwards */
%o local(N=matsize(M)[1]);
%o for (n=1,N, for(j=0,n-1, print1(M[n-j, j+1], ", ") ) );
%o }
%o T(n,k)=
%o {
%o if ( (n==1) || (k==1), return(1) );
%o return( sumdiv(n,d, d*T(d, k-1)) );
%o }
%o M=matrix(15,15,n,k,T(n,k)) /* square array */
%o adu(M) /* sequence */
%Y Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.
%Y Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.
%Y Transposed array: A128119.
%K nonn,tabl,easy
%O 1,5
%A _N. J. A. Sloane_, Nov 19 2009
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