A038998 -id:A038998 - cp's OEIS frontend

A001001 Number of sublattices of index n in generic 3-dimensional lattice.

1, 7, 13, 35, 31, 91, 57, 155, 130, 217, 133, 455, 183, 399, 403, 651, 307, 910, 381, 1085, 741, 931, 553, 2015, 806, 1281, 1210, 1995, 871, 2821, 993, 2667, 1729, 2149, 1767, 4550, 1407, 2667, 2379, 4805, 1723, 5187, 1893, 4655, 4030, 3871, 2257, 8463, 2850, 5642, 3991, 6405, 2863

Offset: 1

N. J. A. Sloane

These sublattices are in 1-1 correspondence with matrices
[a b d]
[0 c e]
[0 0 f]
with acf = n, b = 0..c-1, d = 0..f-1, e = 0..f-1. The sublattice is primitive if gcd(a,b,c,d,e,f) = 1.
Total area of all distinct rectangles whose side lengths are divisors of n, and whose length is an integer multiple of the width. - Wesley Ivan Hurt, Aug 23 2020

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(d), pp. 76 and 113.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
J. Liouville, Théorème concernant les sommes de diviseurs des nombres, Journal de mathématiques pures et appliquées 2e série, tome 2 (1857), p. 56.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [_N. J. A. Sloane_, Mar 14 2009]
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.
Index entries for sequences related to sublattices.

Column 3 of A160870.
Cf. A060983, A064987 (Mobius transform).
Cf. A061256, A226313, A301777.
Primes in this sequence are in A053183.
Cf. A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a001001 n = sum [sum [k * (if k `mod` l == 0 then l else 0) | k <- [1..n], n `mod` k == 0 ] | l <- [1..n]]
a = [ a001001 n | n <- [1..53]]
putStrLn $ concat $ map (++ ", ") (map show a) -- Miles Wilson, Apr 04 2025

nmax := 100:
L12 := [seq(1,i=1..nmax) ];
L27 := [seq(i,i=1..nmax) ];
L290 := [seq(i^2,i=1..nmax) ];
DIRICHLET(L12,L27) ;
DIRICHLET(%,L290) ; # R. J. Mathar, Sep 25 2017

a[n_] := Sum[ d*DivisorSigma[1, d], {d, Divisors[n]}]; Table[ a[n], {n, 1, 42}] (* Jean-François Alcover, Jan 20 2012, after Vladeta Jovovic *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 2}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j);
t=1/prod(j=1,N, eta(x^(j))^j)
t=log(t)
t=serconvol(t,c)
Vec(t)
/* Joerg Arndt, May 03 2008 */

a(n)=sumdiv(n,d, d * sumdiv(d,t, t ) );  /* Joerg Arndt, Oct 07 2012 */

a(n)=sumdivmult(n,d, sigma(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

If n = Product p^m, a(n) = Product (p^(m + 1) - 1)(p^(m + 2) - 1)/(p - 1)(p^2 - 1). Or, a(n) = Sum_{d|n} sigma(n/d)*d^2, Dirichlet convolution of A000290 and A000203.
a(n) = Sum_{d|n} d*sigma(d). - Vladeta Jovovic, Apr 06 2001
Multiplicative with a(p^e) = ((p^(e+1)-1)(p^(e+2)-1))/((p-1)(p^2-1)). - David W. Wilson, Sep 01 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
a(n) = Sum_{d1|n, d2|n, d1|d2} d1*d2. - Wesley Ivan Hurt, Aug 23 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2*zeta(3)/18 = 0.659101... . - Amiram Eldar, Oct 19 2022
G.f.: Sum_{k>=1} Sum {l>=1} k*l^2*x^(k*l - 1)/(1 - x^(k*l)). - Miles Wilson, Apr 04 2025

A038994 Number of sublattices of index n in generic 7-dimensional lattice.

Original entry on oeis.org

1, 127, 1093, 10795, 19531, 138811, 137257, 788035, 896260, 2480437, 1948717, 11798935, 5229043, 17431639, 21347383, 53743987, 25646167, 113825020, 49659541, 210837145, 150021901, 247487059, 154764793, 861322255, 317886556, 664088461, 678468820, 1481689315

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices.

Column 7 of A160870.

Cf. A001001, A038991, A038992, A038993, A038995, A038996, A038997, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 6}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=7.
Multiplicative with a(p^e) = Product_{k=1..6} (p^(e+k)-1)/(p^k-1).
Sum_{k=1..n} a(k) ~ c * n^7, where c = Pi^12*zeta(3)*zeta(5)*zeta(7)/3572100 = 0.325206... . - Amiram Eldar, Oct 19 2022

More terms from Amiram Eldar, Aug 29 2019

A038991 Number of sublattices of index n in generic 4-dimensional lattice.

Original entry on oeis.org

1, 15, 40, 155, 156, 600, 400, 1395, 1210, 2340, 1464, 6200, 2380, 6000, 6240, 11811, 5220, 18150, 7240, 24180, 16000, 21960, 12720, 55800, 20306, 35700, 33880, 62000, 25260, 93600, 30784, 97155, 58560, 78300, 62400, 187550, 52060, 108600, 95200, 217620, 70644, 240000, 81400

Offset: 1

N. J. A. Sloane

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)
M. Baake and U. Grimm, Combinatorial problems of (quasi)crystallography, arXiv:math-ph/0212015, 2002.
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.
Index entries for sequences related to sublattices

Column 4 of A160870.

Cf. A001001, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #&]&]&]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 3}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=4.
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3).
Dirichlet convolution of A000578 and A001001.
Multiplicative with a(p^e) = Product_{k=1..3} (p^(e+k)-1)/(p^k-1).
Sum_{k=1..n} a(k) ~ Pi^6 * Zeta(3) * n^4 / 2160. - Vaclav Kotesovec, Feb 01 2019
Conjectured g.f.: Sum_{k>=1} Sum {l>=1} Sum {m>=1} k*l^2*m^3*x^(k*l*m)/(1 - x^(k*l*m)) (by extension of g.f for A001001). - Miles Wilson, Apr 05 2025

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012

A038992 Number of sublattices of index n in generic 5-dimensional lattice.

Original entry on oeis.org

1, 31, 121, 651, 781, 3751, 2801, 11811, 11011, 24211, 16105, 78771, 30941, 86831, 94501, 200787, 88741, 341341, 137561, 508431, 338921, 499255, 292561, 1429131, 508431, 959171, 925771, 1823451, 732541, 2929531, 954305, 3309747, 1948705, 2750971, 2187581, 7168161, 1926221

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Index entries for sequences related to sublattices.

Column 5 of A160870.

Cf. A001001, A038991, A038993, A038994, A038995, A038996, A038997, A038998, A038999.

a[n_] := DivisorSum[n, #*DivisorSum[#, #*DivisorSum[#, #*DivisorSum[#, # &] &] &] &]; Array[a, 50] (* Jean-François Alcover, Dec 02 2015, after Joerg Arndt *)
f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 4}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n)=sumdiv(n,x, x * sumdiv(x,y, y * sumdiv(y,z, z * sumdiv(z,w, w ) ) ) ); /* Joerg Arndt, Oct 07 2012 */

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=5.
Multiplicative with a(p^e) = Product_{k=1..4} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4). Dirichlet convolution of A038991 with A000583. - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^6*zeta(3)*zeta(5)/2700 = 0.443822... . - Amiram Eldar, Oct 19 2022

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Joerg Arndt, Oct 07 2012

A038993 Number of sublattices of index n in generic 6-dimensional lattice.

Original entry on oeis.org

1, 63, 364, 2667, 3906, 22932, 19608, 97155, 99463, 246078, 177156, 970788, 402234, 1235304, 1421784, 3309747, 1508598, 6266169, 2613660, 10417302, 7137312, 11160828, 6728904, 35364420, 12714681, 25340742, 25095280, 52294536, 21243690, 89572392, 29583456

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and N. Neumarker, A Note on the Relation Between Fixed Point and Orbit Count Sequences, JIS 12 (2009) 09.4.4, Section 3.
Index entries for sequences related to sublattices.

Column 6 of A160870.

Cf. A001001, A038991, A038992, A038994, A038995, A038996, A038997, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 5}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=6.
Multiplicative with a(p^e) = product (p^(e+k)-1)/(p^k-1), k=1..5.
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)*zeta(s-3)*zeta(s-4)*zeta(s-5). Dirichlet convolution of A038992 with A000584. - R. J. Mathar, Mar 31 2011
Sum_{k=1..n} a(k) ~ c * n^6, where c = Pi^12*zeta(3)*zeta(5)/3061800 = 0.376266... . - Amiram Eldar, Oct 19 2022

Offset changed from 0 to 1 by R. J. Mathar, Mar 31 2011

More terms from Amiram Eldar, Aug 29 2019

A038995 Number of sublattices of index n in generic 8-dimensional lattice.

Original entry on oeis.org

1, 255, 3280, 43435, 97656, 836400, 960800, 6347715, 8069620, 24902280, 21435888, 142466800, 67977560, 245004000, 320311680, 866251507, 435984840, 2057753100, 943531280, 4241688360, 3151424000, 5466151440, 3559590240, 20820505200, 7947261556, 17334277800, 18326727760

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices

Column 8 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038996, A038997, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 7}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=8.
Multiplicative with a(p^e) = Product_{k=1..7} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)/43401015000 = 0.285716... . - Amiram Eldar, Oct 19 2022

Offset set to 1 by R. J. Mathar, Mar 01 2011

More terms from Amiram Eldar, Aug 29 2019

A038996 Number of sublattices of index n in generic 9-dimensional lattice.

Original entry on oeis.org

1, 511, 9841, 174251, 488281, 5028751, 6725601, 50955971, 72636421, 249511591, 235794769, 1714804091, 883708281, 3436782111, 4805173321, 13910980083, 7411742281, 37117211131, 17927094321, 85083452531, 66186639441, 120491126959, 81870575521, 501457710611, 198682027181

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices.

Column 9 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038997, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 8}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=9.
Multiplicative with a(p^e) = Product_{k=1..8} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^9, where c = Pi^20*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 38578680000 = 0.254479... . - Amiram Eldar, Oct 19 2022

Offset changed to 1 by R. J. Mathar, Apr 01 2011

More terms from Amiram Eldar, Aug 29 2019

A038997 Number of sublattices of index n in generic 10-dimensional lattice.

Original entry on oeis.org

1, 1023, 29524, 698027, 2441406, 30203052, 47079208, 408345795, 653757313, 2497558338, 2593742460, 20608549148, 11488207654, 48162029784, 72080070744, 222984027123, 125999618778, 668793731199, 340614792100, 1704167305962, 1389966536992, 2653398536580, 1883023236984

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices.

Column 10 of A160870.

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038998, A038999.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 9}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=10.
Multiplicative with a(p^e) = Product_{k=1..9} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k). - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ c * n^10, where c = Pi^30*zeta(3)*zeta(5)*zeta(7)*zeta(9) / 4511535509250000 = 0.229259... . - Amiram Eldar, Oct 19 2022

Offset set to 1 by R. J. Mathar, Apr 01 2011

More terms from Amiram Eldar, Aug 29 2019

A038999 Sublattices of index n in generic 12-dimensional lattice.

Original entry on oeis.org

1, 4095, 265720, 11180715, 61035156, 1088123400, 2306881200, 26167664835, 52955405230, 249938963820, 313842837672, 2970939589800, 1941507093540, 9446678514000, 16218261652320, 57162391576563, 36413889826860, 216852384416850, 122961939948120, 682416684216540

Offset: 1

N. J. A. Sloane

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sublattices.

Cf. A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998.

f[p_, e_] := Product[(p^(e + k) - 1)/(p^k - 1), {k, 1, 11}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

f(Q, n) = Sum_{d|n} d*f(Q-1, d); here Q=12.
Multiplicative with a(p^e) = Product_{k=1..11} (p^(e+k)-1)/(p^k-1).
Dirichlet g.f.: Product_{k=0..Q-1} zeta(s-k).
Sum_{k=1..n} a(k) ~ c * n^12, where c = Pi^42*zeta(3)*zeta(5)*zeta(7)*zeta(9)*zeta(11)/3456808210410967912500000 = 0.191191... . - Amiram Eldar, Oct 19 2022

Offset set to 1 by R. J. Mathar, Apr 01 2011

More terms from Amiram Eldar, Aug 29 2019

A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, May 09 2007

Differs from sum of divisors of m^(n-1) in 4th column!

			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,

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

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.

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.

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 *)

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)))

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

Edited by Charles R Greathouse IV, Oct 28 2009

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A001001 Number of sublattices of index n in generic 3-dimensional lattice.

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A038994 Number of sublattices of index n in generic 7-dimensional lattice.

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A038991 Number of sublattices of index n in generic 4-dimensional lattice.

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A038992 Number of sublattices of index n in generic 5-dimensional lattice.

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A038993 Number of sublattices of index n in generic 6-dimensional lattice.

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A038995 Number of sublattices of index n in generic 8-dimensional lattice.

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A038996 Number of sublattices of index n in generic 9-dimensional lattice.

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