A004525 -id:A004525 - cp's OEIS frontend

A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

1, 2, 3, 5, 4, 7, 5, 10, 8, 10, 7, 17, 8, 13, 14, 19, 10, 21, 11, 24, 18, 19, 13, 35, 17, 22, 22, 31, 16, 38, 17, 36, 26, 28, 26, 50, 20, 31, 30, 50, 22, 50, 23, 45, 42, 37, 25, 69, 30, 48, 38, 52, 28, 62, 38, 65, 42, 46, 31, 90, 32, 49, 55, 69, 44, 74, 35, 66, 50, 74

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

The centered rectangular lattice has symmetry group c2mm, or cmm. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] [see Table 8].
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2.]
Index entries for sequences related to sublattices

Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525.

a060594[n_] := Switch[Mod[n, 8], 2|6, 2^(PrimeNu[n] - 1), 1|3|4|5|7, 2^PrimeNu[n], 0, 2^(PrimeNu[n] + 1)];
a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];
a[n_] := (DivisorSigma[1, n] + a145390[n])/2;
Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A060594(n/m^2) + A157223(n/m^2) = A145390(n) + Sum_{ m: m^2|n } A157223(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A004525(d+1). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Mar 12 2018

New name from Andrey Zabolotskiy, Jan 19 2022

A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

Original entry on oeis.org

0, -1, -2, -3, -4, -10, -20, -35, -56, -126, -252, -462, -792, -1716, -3432, -6435, -11440, -24310, -48620, -92378, -167960, -352716, -705432, -1352078, -2496144, -5200300, -10400600, -20058300, -37442160, -77558760, -155117520, -300540195

Offset: 1

Terence Tao, Jul 09 2012

sign

An m-downset is a set of subsets of 1..m such that if S is in the set, so are all subsets of S. The Euler characteristic of a downset is the number of sets in the downset with an even cardinality, minus the number with an odd cardinality.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Terry Tao, Optimal bounds for an alternating sum on a downset, 2012.

Cf. A000108, A001405, A004525, A007318, A212831, A214282.

a214283 1 = 0
a214283 n = - a007318 (n - 1) (a004525 n)
-- Reinhard Zumkeller, Jul 14 2012

A212831[n_] := (1/4)*((-(1+(-1)^n))*(-1+(-1)^Floor[n/2]) - (-3+(-1)^n)*n); a[n_] := -CatalanNumber[Floor[(n-1)/2]]*A212831[n-1]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Nov 06 2012, after Paul Curtz *)

a(n)=-binomial(n-1,if(n%2,if(n%4==3,n-1,n+1),n)/2) \\ Charles R Greathouse IV, Jul 10 2012

from math import comb
def A214283(n): return -comb(n-1,(n>>1)|(n&1)) # Chai Wah Wu, Jan 31 2024

a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),
a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),
a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).
a(n) = A214282(n) - A001405(n). - Reinhard Zumkeller, Jul 14 2012
For n > 1: a(n) = - A007318(n-1, A004525(n)). - Reinhard Zumkeller, Jul 14 2012
a(n+1) = -A000108(n/2) * A212831(n). - Paul Curtz, Nov 04 2012

A091573 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.

Original entry on oeis.org

7, 12, 17, 24, 31, 36, 41, 48, 55, 60, 65, 72, 79, 84, 89, 96, 103, 108, 113, 120, 127, 132, 137, 144, 151, 156, 161, 168, 175, 180, 185, 192, 199, 204, 209, 216, 223, 228, 233, 240, 247, 252, 257, 264, 271, 276, 281, 288, 295, 300, 305, 312, 319, 324, 329

Offset: 0

Paul Boddington, Jan 22 2004

I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, Vol. 44, Number 5.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A004525, A091571, A091572, A091574, A091575, A091576, A091577.

CoefficientList[ Series[ (7 - 2x + 7x^2) / (1 - 2x + 2x^2 - 2x^3 + x^4), {x, 0, 49}], x] (* Jean-François Alcover, Dec 02 2011 *)

a(n) = (12+(-I)^n+I^n+12*n)/2 \\ Colin Barker, Oct 18 2015

Vec((7-2*x+7*x^2)/((1+x^2)*(1-x)^2) + O(x^100)) \\ Colin Barker, Oct 18 2015

a(n) = if(n%2 == 1, 6*n+6, if(n%4 == 0, 6*n+7, 6*n+5));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 18 2015

a(n) = 6*n+6 (n odd), 6*n+7 (n==0 (mod 4)), 6*n+5 (n==2 (mod 4)).
G.f.: (7-2*x+7*x^2) / ((1+x^2)*(1-x)^2).
From Colin Barker, Oct 18 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>3.
a(n) = (12+(-i)^n+i^n+12*n)/2 where i = sqrt(-1).
(End)

G.f. corrected by Colin Barker, Oct 18 2015

A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 4, 3, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 5, 4, 5, 4, 3, 2, 1, 1, 1

Offset: 0

Seiichi Manyama, Jul 10 2018

			T(8,4) = 3.
    *                             *
   / \                           / \
  *---*   *     *---*---*       *---*
   \ / \ / \     \ / \ / \     / \ / \
    *---*---*     *---*---*   *---*---*
     \ / \ /       \ / \ /     \ / \ /
      *---*         *---*       *---*
       \ /           \ /         \ /
        *             *           *
Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 1, 1, 1;
  0, 0, 1, 1, 1, 1;
  0, 0, 1, 1, 1, 1,  1;
  0, 0, 1, 1, 2, 1,  1,  1;
  0, 0, 1, 1, 3, 2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  3,  2,  1,  1,  1;
  0, 0, 1, 1, 4, 3,  4,  3,  2,  1,  1, 1;
  0, 0, 1, 1, 5, 4,  5,  4,  3,  2,  1, 1, 1;
  0, 0, 1, 1, 5, 5,  6,  5,  4,  3,  2, 1, 1, 1;
  0, 0, 1, 1, 5, 5,  8,  6,  5,  4,  3, 2, 1, 1, 1;
  0, 0, 1, 1, 6, 5, 10,  8,  7,  5,  4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 6, 11, 10, 10,  7,  5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 13, 11, 12, 10,  7, 5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A006950.
Sums of even columns give A059777.
Cf. A004525, A000933, A089597, A014670, A316718, A316719, A316720, A316721, A316722.
Cf. A072233.

For m >= 0,
Sum_{n>=2m}   T(n,2m)  *x^n = x^(2m)   * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).
Sum_{n>=2m+1} T(n,2m+1)*x^n = x^(2m+1) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).

A316718 Expansion of Product_{k=1..6} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 34, 41, 52, 65, 78, 93, 113, 137, 162, 189, 224, 266, 308, 355, 414, 480, 549, 626, 717, 820, 928, 1045, 1183, 1337, 1496, 1670, 1871, 2091, 2321, 2571, 2853, 3161, 3484, 3830, 4218, 4640, 5078, 5549, 6072, 6633, 7219

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), this sequence (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 6, (1+x^(2*k-1))/(1-x^(2*k))))

A316719 Expansion of Product_{k=1..7} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 54, 68, 83, 100, 122, 149, 179, 212, 253, 303, 357, 417, 490, 575, 668, 772, 893, 1033, 1187, 1356, 1551, 1773, 2015, 2281, 2583, 2922, 3291, 3695, 4147, 4650, 5197, 5791, 6450, 7179, 7966, 8818, 9757, 10785, 11893

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), this sequence (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

nmax=60; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 7}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, 7, (1+x^(2*k-1))/(1-x^(2*k))))

A316720 Expansion of Product_{k=1..8} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 85, 103, 127, 156, 188, 224, 270, 326, 386, 454, 539, 638, 746, 869, 1016, 1186, 1372, 1581, 1827, 2108, 2415, 2758, 3156, 3605, 4094, 4639, 5261, 5956, 6715, 7553, 8499, 9552, 10694, 11950, 13357, 14908, 16589

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), this sequence (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 8, (1+x^(2*k-1))/(1-x^(2*k))))

A316721 Expansion of Product_{k=1..9} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 129, 159, 193, 231, 279, 338, 403, 477, 568, 675, 795, 932, 1094, 1284, 1497, 1736, 2016, 2340, 2700, 3105, 3573, 4106, 4699, 5363, 6118, 6972, 7921, 8974, 10163, 11500, 12974, 14606, 16435, 18471

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), this sequence (b=9), A316722 (b=10).

Cf. A316675.

nmax=50; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 9}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, 9, (1+x^(2*k-1))/(1-x^(2*k))))

A316722 Expansion of Product_{k=1..10} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 195, 234, 284, 345, 412, 489, 585, 698, 824, 969, 1143, 1347, 1575, 1834, 2141, 2496, 2891, 3339, 3862, 4460, 5125, 5876, 6740, 7720, 8810, 10031, 11423, 12993, 14730, 16669, 18862, 21315

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), this sequence (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 10, (1+x^(2*k-1))/(1-x^(2*k))))

A134314 First differences of A134429.

Original entry on oeis.org

-8, 8, -8, 16, -24, 24, -24, 32, -40, 40, -40, 48, -56, 56, -56, 64, -72, 72, -72, 80, -88, 88, -88, 96, -104, 104, -104, 112, -120, 120, -120, 128, -136, 136, -136, 144, -152, 152, -152, 160, -168, 168, -168, 176, -184, 184, -184, 192, -200, 200, -200, 208

Offset: 0

Paul Curtz, Jan 30 2008

Index entries for linear recurrences with constant coefficients, signature (-2,-2,-2,-1).

A134429 := proc(n) npr := floor(n/4) ; if (n mod 4 =0) or (n mod 4 = 2) then 8*npr+3 ; else -8*npr-5 ; fi; end: A134314 := proc(n) A134429(n+1)-A134429(n) ; end: seq(A134314(n),n=0..80) ; # R. J. Mathar, Feb 07 2009

LinearRecurrence[{-2, -2, -2, -1}, {-8, 8, -8, 16}, 52] (* Jean-François Alcover, Mar 31 2020 *)

From R. J. Mathar, Feb 07 2009: (Start)
a(n)= -2*a(n-1)-2*a(n-2)-2*a(n-3)-a(n-4) = -8*(-1)^n*A004525(n+1).
G.f.: -8*(1+x+x^2)/((1+x^2)*(1+x)^2). (End)

Edited by N. J. A. Sloane, Mar 23 2008

More terms from R. J. Mathar, Feb 07 2009

cp's OEIS Frontend

A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

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A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

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A091573 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type E_6.

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A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

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A316718 Expansion of Product_{k=1..6} (1+x^(2*k-1))/(1-x^(2*k)).

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A316719 Expansion of Product_{k=1..7} (1+x^(2*k-1))/(1-x^(2*k)).

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A316720 Expansion of Product_{k=1..8} (1+x^(2*k-1))/(1-x^(2*k)).

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A316721 Expansion of Product_{k=1..9} (1+x^(2*k-1))/(1-x^(2*k)).

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A316718 Expansion of Product_{k=1..6} (1+x^(2k-1))/(1-x^(2k)).

A316719 Expansion of Product_{k=1..7} (1+x^(2k-1))/(1-x^(2k)).

A316720 Expansion of Product_{k=1..8} (1+x^(2k-1))/(1-x^(2k)).

A316721 Expansion of Product_{k=1..9} (1+x^(2k-1))/(1-x^(2k)).

A316722 Expansion of Product_{k=1..10} (1+x^(2k-1))/(1-x^(2k)).