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 A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1). 25
 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, 1, 8, 91, 156, 651, 364, 127, 1, 1, 15, 57, 600, 781, 2667, 1093, 255, 1, 1, 13, 155, 400, 3751, 3906, 10795, 3280, 511, 1, 1, 18, 130, 1395, 2801, 22932, 19531, 43435, 9841, 1023, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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. J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006. Yi Ming Zou, Gaussian binomials and the number of sublattices, Acta Cryst. A62 (2006) 409-410. FORMULA T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1). From Álvar Ibeas, Oct 31 2015: (Start) T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1). T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p). (End) EXAMPLE Array begins: 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,... 1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,... 1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,... 1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,... 1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,... ... MATHEMATICA 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 *) PROG (PARI): adu(M)= { /* Read by AntiDiagonals, Upwards */     local(N=matsize(M)[1]);     for (n=1, N, for(j=0, n-1, print1(M[n-j, j+1], ", ") ) ); } T(n, k)= {     if ( (n==1) || (k==1), return(1) );     return( sumdiv(n, d, d*T(d, k-1)) ); } M=matrix(15, 15, n, k, T(n, k)) /* square array */ adu(M) /* sequence */ CROSSREFS Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997. Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096. Transposed array: A128119. Sequence in context: A208344 A209172 A263950 * A345279 A342447 A025255 Adjacent sequences:  A160867 A160868 A160869 * A160871 A160872 A160873 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Nov 19 2009 STATUS approved

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Last modified September 22 17:39 EDT 2021. Contains 347607 sequences. (Running on oeis4.)