A016152 -id:A016152 - cp's OEIS frontend

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640

Offset: 1

Álvar Ibeas, Oct 30 2015

All the enumerated lattices have full rank k, since the quotient group is finite.
For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.
Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].
T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).
Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
Columns are multiplicative functions.

			There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.
Array begins:
      k=1    k=2    k=3    k=4    k=5    k=6
n=1     1      1      1      1      1      1
n=2     1      3      7     15     31     63
n=3     1      4     13     40    121    364
n=4     1      6     28    120    496   2016
n=5     1      6     31    156    781   3906
n=6     1     12     91    600   3751  22932

Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.

Álvar Ibeas, First 100 antidiagonals, flattened
Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
Jin Ho Kwak and Jaeun 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. [Examples 2 and 3]

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
Main diagonal gives A344210.
Cf. A000203, A008683, A160870.

f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 08 2022 *)

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^k). - Ridouane Oudra, Apr 03 2025

A016140 Expansion of 1/((1-3x)(1-8*x)).

Original entry on oeis.org

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

A075499 Stirling2 triangle with scaled diagonals (powers of 4).

Original entry on oeis.org

1, 4, 1, 16, 12, 1, 64, 112, 24, 1, 256, 960, 400, 40, 1, 1024, 7936, 5760, 1040, 60, 1, 4096, 64512, 77056, 22400, 2240, 84, 1, 16384, 520192, 989184, 435456, 67200, 4256, 112, 1, 65536, 4177920, 12390400, 7956480, 1779456, 169344, 7392, 144, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(4*z) - 1)*x/4) - 1
Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			[1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     4      1
*    16     12      1
*    64    112     24      1
*   256    960    400     40     1
*  1024   7936   5760   1040    60    1
*  4096  64512  77056  22400  2240   84   1
* 16384 520192 989184 435456 67200 4256 112 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.

Cf. A075498, A075500.

Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
# Adds a column 1,0,0,... at the left side of the triangle.
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
inverse_bell_matrix(multifact_4_4, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (4^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 4m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-4k*x), m >= 1.
E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1.

A092237 Maximum number of intercalates in a Latin square of order n.

Original entry on oeis.org

0, 1, 0, 12, 4, 27, 42, 112, 72

Offset: 1

Richard Bean, Feb 17 2004

An intercalate is a 2 X 2 subsquare of a Latin square. a(10) >= 125, a(11) >= 172, a(12) >= 324.
a(13) >= 208, a(14) >= 391, a(15) >= 630, a(16) >= 960, a(17) >= 736, a(18) >= 729, a(19) >= 472, a(20) >= 1500, a(21) >= 884, a(22) >= 1497, a(23) >= 983, a(24) >= 1872, a(25) >= 1700, a(26) >= 2197, a(27) >= 648, a(28) >= 2940. - Eduard I. Vatutin, Mar 02 2025
If, in theory, all unordered pairs of rows and columns form intercalate in their intersection, total number of intercalates will be (n*(n-1))^2, so a(n) <= (n*(n-1))^2, a(n) is asymptotically less than O(n^4). In practice a(n) << (n*(n-1))^2. - Eduard I. Vatutin, Mar 11 2025
a(2^n-1) = 42*A006096(n) for n > 2. - Eduard I. Vatutin, Apr 23 2025
Due to existence of the pine Latin squares for even orders N=2n, a(2n) >= A383368(n). Pine Latin squares exist for all even orders, so a(N) >= (2k)^2 * (2k + 1) for N=4k and a(N) >= (2k+1)^3 for N=4k+2. Therefore, asymptotically maximum number of intercalates in Latin squares of even orders N greater or equal than o(k1*N^3), where k1 = 1/8. - Eduard I. Vatutin, Apr 30 2025
From Eduard I. Vatutin, Jul 01 2025: (Start)
Table showing minimums and maximums:
  order                      | 1 2 3  4 5  6  7   8   9    10    11    12    13    14    15    16    17    18    19
  min number of intercalates | 0 1 0  4 0  0  0   0   0     0     0     0     0     0     0     0     0     0     0
  max number of intercalates | 0 1 0 12 4 27 42 112  72 >=125 >=172 >=324 >=208 >=391 >=630 >=960 >=736 >=729 >=472 (this sequence)
.
  order                      |      20    21     22    23     24     25     26    27     28 29
  min number of intercalates |       0     0      0     0      0      0   <=15     0    <=1  0
  max number of intercalates |  >=1500 >=884 >=1497 >=983 >=1872 >=1700 >=2197 >=648 >=2940  ? (this sequence)
(End)

I. Wanless, Private communication, 2003.

R. Bean, Critical sets in Latin squares and associated structures, Ph.D. Thesis, The University of Queensland, 2001.
K. Heinrich and W. Wallis, The maximum number of intercalates in a Latin square, Combinatorial Math. VIII, Proc. 8th Australian Conf. Combinatorics, 1980, 221-233.
Eduard I. Vatutin, Proving list, minimum number of intercalates (best known examples).
Eduard I. Vatutin, Proving list, maximum number of intercalates (best known examples).
Eduard I. Vatutin, About interconnection between maximum number of intercalates in Latin squares of order N=2^n-1 and Gaussian binomial coefficients [n,3] for q=2 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006096, A016152, A091323, A090741, A383368.

If n is a power of 2, a(n) = n^2*(n-1)/4 = A016152(log2(n)); if n is one less than a power of 2, a(n) = n*(n-1)*(n-3)/4 = A006096(log2(n+1))*42. - updated by Eduard I. Vatutin, Jun 28 2025

A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

Original entry on oeis.org

2, 24, 224, 1920, 15872, 129024, 1040384, 8355840, 66977792, 536346624, 4292870144, 34351349760, 274844352512, 2198889037824, 17591649173504, 140735340871680, 1125891316908032, 9007164895002624, 72057456598974464, 576460202547609600

Offset: 1

Omar E. Pol, Nov 06 2008

a(n) is the number whose binary representation is A138119(n).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A138119.

Cf. A016152. - Omar E. Pol, Nov 13 2008

List([1..20], n-> 2^(2*n-1)*(2^n -1)); # G. C. Greubel, Jan 12 2020

[2^(2*n-1)*(2^n -1): n in [1..20]]; // G. C. Greubel, Jan 12 2020

seq(2^(2*n-1)*(2^n -1), n=1..20); # G. C. Greubel, Jan 12 2020

Table[FromDigits[Join[Table[1, {n}], Table[0, {2n - 1}]], 2], {n, 1, 20}] (* Stefan Steinerberger, Nov 11 2008 *)

vector(20, n, 2^(2*n-1)*(2^n -1)) \\ G. C. Greubel, Jan 12 2020

def a(n): return ((1 << n) - 1) << (2*n-1)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 24 2021

[2^(2*n-1)*(2^n -1) for n in (1..20)] # G. C. Greubel, Jan 12 2020

a(n) = 2^(2*n-1)*(2^n -1) = A081294(n)*A000225(n). - R. J. Mathar, Nov 09 2008
a(n) = 2*A016152(n). - Omar E. Pol, Nov 13 2008
From Colin Barker, Nov 04 2012: (Start)
a(n) = 12*a(n-1) - 32*a(n-2).
G.f.: 2*x/((1-4*x)*(1-8*x)). (End)

Extended by R. J. Mathar and Stefan Steinerberger, Nov 09 2008

A016290 Expansion of g.f. 1/((1-2x)(1-4x)(1-8*x)).

Original entry on oeis.org

1, 14, 140, 1240, 10416, 85344, 690880, 5559680, 44608256, 357389824, 2861214720, 22898104320, 183218384896, 1465881288704, 11727587164160, 93822844764160, 750591347982336, 6004765143465984, 48038258586419200, 384306618446643200, 3074455146595352576, 24595649968853745664

Offset: 0

N. J. A. Sloane

a(n) is the number of quads in the EvenQuads-2^{n+2} deck. - Tanya Khovanova and MIT PRIMES STEP senior group, Jul 02 2023

T. D. Noe, Table of n, a(n) for n=0..100
Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, and Raphael Walker, How many cards should you lay out in a game of EvenQuads? A detailed study of 2-caps in AG(n,2), arXiv:2212.05353 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Cf. A006516, A016152, A160873.

[(2^n-6*4^n+8*8^n)/3 : n in [0..20]]; // Wesley Ivan Hurt, Jul 07 2014

[seq(GBC(n+1,3,2)-GBC(n,3,2), n=2..30)]; # produces A016290 (cf. A006516).
seq((2^n-6*4^n+8*8^n)/3, n=0..20);
seq(binomial(2^n,3)/4, n=2..20); # Zerinvary Lajos, Feb 22 2008

CoefficientList[Series[1/((1-2x)(1-4x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[{14,-56,64},{1,14,140},30] (* Harvey P. Dale, Jul 23 2011 *)

G.f.: 1/((1-2*x)*(1-4*x)*(1-8*x)).
Difference of Gaussian binomial coefficients [ n+1, 3 ] - [ n, 3 ] (n >= 2).
a(n) = (2^n - 6*4^n + 8*8^n)/3. - James R. Buddenhagen, Dec 14 2003
a(n) = Sum_{0<=i,j,k,<=n; i+j+k=n} 2^i*4^j*8^k. - Hieronymus Fischer, Jun 25 2007
From Vincenzo Librandi, Mar 15 2011: (Start)
a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3) for n >= 3.
a(n) = 12*a(n-1) - 32*a(n-2) + 2^n with a(0)=1, a(1)=14. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(8*exp(6*x) - 6*exp(2*x) + 1)/3.
a(n) = A160873(n+2)/3. (End)

A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).

Original entry on oeis.org

1, 24, 400, 5760, 77056, 989184, 12390400, 152862720, 1867841536, 22682271744, 274333696000, 3309180026880, 39847582498816, 479270434504704, 5760041038643200, 69190860134154240, 830853267268304896, 9974742789667160064, 119732942204305408000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (24,-176,384).

Third column of triangle A075499.

Cf. A016152, A075907.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-8*x)*(1-12*x)))); /* or */ I:=[1, 24, 400]; [n le 3 select I[n] else 24*Self(n-1)-176*Self(n-2)+384*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [24, -176, 384][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # Alois P. Heinz, Jul 03 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 8 x) (1 - 12 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{24,-176,384},{1,24,400},20] (* Harvey P. Dale, Jul 18 2020 *)

Vec(1/((1-4*x)*(1-8*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = (4^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = (4^n - 8*8^n + 9*12^n)/2.
G.f.: 1/((1-4*x)*(1-8*x)*(1-12*x)).
E.g.f.: (d^3/dx^3)((((exp(4*x)-1)/4)^3)/3!) = (exp(4*x) - 8*exp(8*x) + 9*exp(12*x))/2.
a(0)=1, a(1)=24, a(2)=400; for n > 2, a(n) = 24*a(n-1) - 176*a(n-2) + 384*a(n-3). - Vincenzo Librandi, Jul 03 2013
a(n) = 30*a(n-1) - 96*a(n-2) + 4^n. - Vincenzo Librandi, Jul 03 2013

A059409 a(n) = 4^n * (2^n - 1).

Original entry on oeis.org

0, 4, 48, 448, 3840, 31744, 258048, 2080768, 16711680, 133955584, 1072693248, 8585740288, 68702699520, 549688705024, 4397778075648, 35183298347008, 281470681743360, 2251782633816064, 18014329790005248, 144114913197948928, 1152920405095219200

Offset: 0

N. J. A. Sloane, Alford Arnold, Jan 30 2001

Jordan's totient functions are described more fully in A059379 and A059380; for example, J_1(n) is Euler's totient function and J_2(n) the Moebius transform of squares.

			(4,48,448,3840,...) = (8,64,512,4096,...) - (2,12,56,240,...) - (1,3,7,15,...) - (1,1,1,1,...)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A059379, A059380, A016152.
Cf. A024023, A000225, A000012.
Cf. A065442.

List([0..100], n->4^n * (2^n - 1)); # Muniru A Asiru, Jan 29 2018

[4^n*(2^n - 1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

seq(4^n * (2^n - 1), n=0..20); # Muniru A Asiru, Jan 29 2018

Table[4^n*(2^n - 1), {n,0,30}] (* G. C. Greubel, Jan 29 2018 *)
LinearRecurrence[{12,-32},{0,4},20] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = { 4^n*(2^n - 1) } \\ Harry J. Smith, Jun 26 2009

Equals J_n(8) (see A059379).
J_n(8) = 8^n - A024023(n) - A000225(n) - A000012(n).
a(n) = 4*A016152(n).
G.f.: 4*x / ( (8*x-1)*(4*x-1) ). - R. J. Mathar, Nov 23 2018
Sum_{n>0} 1/a(n) = E - 4/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A291779(A008585(n)) = A045991(A000079(n)). - Mathew Englander, Feb 08 2024

A147816 Concatenation of n digits 1 and 2(n-1) digits 0.

Original entry on oeis.org

1, 1100, 1110000, 1111000000, 1111100000000, 1111110000000000, 1111111000000000000, 1111111100000000000000, 1111111110000000000000000, 1111111111000000000000000000, 1111111111100000000000000000000, 1111111111110000000000000000000000

Offset: 1

Omar E. Pol, Nov 13 2008

a(n) is also A016152(n) written in base 2.

			n ...... a(n)
1 ....... 1
2 ...... 1100
3 ..... 1110000
4 .... 1111000000
5 ... 1111100000000

Paolo Xausa, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A000533, A016152, A135577, A138119, A138120, A138144, A138145, A138146, A138721, A138826, A147757, A147759.

Array[(10^#-1)*10^(2*#-2)/9 &, 20] (* or *)
LinearRecurrence[{1100, -100000}, {1, 1100}, 20] (* Paolo Xausa, Feb 27 2024 *)

Vec(x/((100*x-1)*(1000*x-1))  + O(x^100)) \\ Colin Barker, Sep 16 2013

a(n) = A138119(n)/10.

a(n) = 1100*a(n-1)-100000*a(n-2). G.f.: x / ((100*x-1)*(1000*x-1)). - Colin Barker, Sep 16 2013

cp's OEIS Frontend

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

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A016140 Expansion of 1/((1-3*x)*(1-8*x)).

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A075499 Stirling2 triangle with scaled diagonals (powers of 4).

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A092237 Maximum number of intercalates in a Latin square of order n.

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A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

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A016290 Expansion of g.f. 1/((1-2*x)*(1-4*x)*(1-8*x)).

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A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).

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A016140 Expansion of 1/((1-3x)(1-8*x)).

A016290 Expansion of g.f. 1/((1-2x)(1-4x)(1-8*x)).