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%I #37 Dec 10 2018 08:54:03
%S 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,
%T 1,127,364,651,156,91,8,1,1,255,1093,2667,781,600,57,15,1,1,511,3280,
%U 10795,3906,3751,400,155,13,1,1,1023,9841,43435,19531,22932,2801,1395,130,18,1
%N Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.
%C Differs from sum of divisors of m^(n-1) in 4th column!
%D Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]
%H Álvar Ibeas, <a href="/A128119/b128119.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 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.
%F Dirichlet g.f. of n-th row: Product_{i=0..n-1} zeta(s-i).
%F If m is squarefree, T(n,m) = A000203(m^(n-1)). - _Álvar Ibeas_, Jan 17 2015
%F T(n, Product(p^e)) = Product(Gaussian_poly[e+n-1, e]_p). - _Álvar Ibeas_, Oct 31 2015
%e Array starts:
%e 1,1,1,1,1,1,1,1,1,
%e 1,3,4,7,6,12,8,15,13,
%e 1,7,13,35,31,91,57,155,130,
%e 1,15,40,155,156,600,400,1395,1210,
%e 1,31,121,651,781,3751,2801,11811,11011,
%e 1,63,364,2667,3906,22932,19608,97155,99463,
%e 1,127,1093,10795,19531,138811,137257,788035,896260,
%e 1,255,3280,43435,97656,836400,960800,6347715,8069620,
%t 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}]];
%t Table[T[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 10 2018 *)
%o (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)))
%Y Rows include A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
%Y Columns include A000225, A003462, A006095, A003463, A160869, A023000, A006096, A006100, A046915.
%Y Transposed array is A160870.
%K nonn,tabl
%O 1,5
%A _Ralf Stephan_, May 09 2007
%E Edited by _Charles R Greathouse IV_, Oct 28 2009
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