A038994 -id:A038994 - cp's OEIS frontend

A068023 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=6.

1, 127, 1093, 10795, 19531, 164809, 137257, 788035, 896260, 2745247, 1948717, 15172249, 5229043, 18728221, 22858948, 53743987, 25646167, 142560946, 49659541, 244930015, 157475284, 258931921, 154764793, 1151073625, 317886556

Offset: 1

Vladeta Jovovic, Feb 08 2002

nonn

Cf. A067692, A068020-A068022, A068024-A068027, A000203, A001157-A001160, A013954.

CIP6 = CycleIndexPolynomial[SymmetricGroup[6], Array[x, 6]]; a[n_] := CIP6 /. x[k_] -> DivisorSigma[k, n]; Array[a, 25] (* Jean-François Alcover, Nov 04 2016 *)

1/6!*(sigma[1](n)^6 + 15*sigma[1](n)^4*sigma[2](n) + 40*sigma[1](n)^3*sigma[3](n) + 45*sigma[1](n)^2*sigma[2](n)^2 + 90*sigma[1](n)^2*sigma[4](n) + 120*sigma[1](n)*sigma[2](n)*sigma[3](n) + 15*sigma[2](n)^3 + 144*sigma[1](n)*sigma[5](n) + 90*sigma[2](n)*sigma[4](n) + 40*sigma[3](n)^2 + 120*sigma[6](n)).

Agrees with A038994 at n = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23... - Ralf Stephan, Mar 09 2004

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

cp's OEIS Frontend

A068023 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=6.

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

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