A008413 -id:A008413 - cp's OEIS frontend

A113413 A Riordan array of coordination sequences.

1, 2, 1, 2, 4, 1, 2, 8, 6, 1, 2, 12, 18, 8, 1, 2, 16, 38, 32, 10, 1, 2, 20, 66, 88, 50, 12, 1, 2, 24, 102, 192, 170, 72, 14, 1, 2, 28, 146, 360, 450, 292, 98, 16, 1, 2, 32, 198, 608, 1002, 912, 462, 128, 18, 1, 2, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 1, 2, 40, 326

Offset: 0

Paul Barry, Oct 29 2005

Columns include A040000, A008574, A005899, A008412, A008413, A008414. Row sums are A078057(n)=A001333(n+1). Diagonal sums are A001590(n+3). Reverse of A035607. Signed version is A080246. Inverse is A080245.
For another version see A122542. - Philippe Deléham, Oct 15 2006
T(n,k) is the number of length n words on alphabet {0,1,2} with no two consecutive 1's and no two consecutive 2's and having exactly k 0's. - Geoffrey Critzer, Jun 11 2015
From Eric W. Weisstein, Feb 17 2016: (Start)
Triangle of coefficients (from low to high degree) of x^-n * vertex cover polynomial of the n-ladder graph P_2 \square p_n:
  Psi_{L_1}: x*(2 + x) -> {2, 1}
  Psi_{L_2}: x^2*(2 + 4 x + x^2) -> {2, 4, 1}
  Psi_{L_3}: x^3*(2 + 8 x + 6 x^2 + x^3) -> {2, 8, 6, 1}
(End)
Let c(n, k), n > 0, be multiplicative sequences for some fixed integer k >= 0 with c(p^e, k) = T(e+k, k) for prime p and e >= 0. Then we have Dirichlet g.f.: Sum_{n>0} c(n, k) / n^s = zeta(s)^(2*k+2) / zeta(2*s)^(k+1). Examples: For k = 0 see A034444 and for k = 1 see A322328. Dirichlet convolution of c(n, k) and lambda(n) is Dirichlet inverse of c(n, k). - Werner Schulte, Oct 31 2022

			Triangle begins
  1;
  2,  1;
  2,  4,  1;
  2,  8,  6,  1;
  2, 12, 18,  8,  1;
  2, 16, 38, 32, 10,  1;
  2, 20, 66, 88, 50, 12,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
P. Holub, M. Miller, H. Perez-Roses, and J. Ryan, Degree diameter problem on honeycomb networks, Disc. Appl. Math. 179 (2014) 139-151.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 2.
Mirka Miller, Hebert Perez-Roses, and Joe Ryan, The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh, arXiv preprint arXiv:1203.4069 [math.CO], 2012.

Other versions: A035607, A119800, A122542, A266213.

nn = 10; Map[Select[#, # > 0 &] &, CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, nn}], {x, y}]] // Grid (* Geoffrey Critzer, Jun 11 2015 *)
CoefficientList[CoefficientList[Series[1/(1 - 2 x/(1 + x) - y x), {x, 0, 10}, {y, 0, 10}], x], y] (* Eric W. Weisstein, Feb 17 2016 *)

T = lambda n,k : binomial(n, k)*hypergeometric([-k-1, k-n], [-n], -1).simplify_hypergeometric()
A113413 = lambda n,k : 1 if n==0 and k==0 else T(n, k)
for n in (0..12): print([A113413(n,k) for k in (0..n)]) # Peter Luschny, Sep 17 2014 and Mar 16 2016

# Alternatively:
def A113413_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)+2*sum(prec(n-i,k-1) for i in (2..n-k+1))
    return [prec(n, k) for k in (1..n)]
for n in (1..10): print(A113413_row(n)) # Peter Luschny, Mar 16 2016

From Paul Barry, Nov 13 2005: (Start)
Riordan array ((1+x)/(1-x), x(1+x)/(1-x)).
T(n, k) = Sum_{i=0..n-k} C(k+1, i)*C(n-i, k).
T(n, k) = Sum_{j=0..n-k} C(k+j, j)*C(k+1, n-k-j).
T(n, k) = D(n, k) + D(n-1, k) where D(n, k) = Sum_{j=0..n-k} C(n-k, j)*C(k, j)*2^j = A008288(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-1).
T(n, k) = Sum_{j=0..n} C(floor((n+j)/2), k)*C(k, floor((n-j)/2)). (End)
T(n, k) = C(n, k)*hypergeometric([-k-1, k-n], [-n], -1). - Peter Luschny, Sep 17 2014
T(n, k) = (Sum_{i=2..k+2} A137513(k+2, i) * (n-k)^(i-2)) / (k!) for 0 <= k < n (conjectured). - Werner Schulte, Oct 31 2022

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

Original entry on oeis.org

4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22

Offset: 1

Thomas Wieder, Jul 30 2006, Aug 06 2006

			The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
The main diagonal seems to be A050146.
Square array A(n,k) begins:
   4,   8,   12,   16,    20,    24,     28,     32,      36, ...
   6,  18,   38,   66,   102,   146,    198,    258,     326, ...
   8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
  10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
  12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
  14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
  16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
  18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
  20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Other versions: A035607, A113413, A122542, A266213.

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.

A:= proc(m, n)  option remember;
      `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Apr 21 2012

A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.

Offset and typos corrected by Alois P. Heinz, Apr 21 2012

A008416 Coordination sequence for 8-dimensional cubic lattice.

Original entry on oeis.org

1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120, 425000576

Offset: 0

N. J. A. Sloane

Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

Cf. A113413, A122542.

CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* Michael De Vlieger, Dec 18 2017 *)

G.f.: ((1+x)/(1-x))^8.
a(n) = A008415(n) + A008415(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017
n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 20, 102, 360, 1002, 2364, 4942, 9424, 16722, 28004, 44726, 68664, 101946, 147084, 207006, 285088, 385186, 511668, 669446, 864008, 1101450, 1388508, 1732590, 2141808, 2625010, 3191812, 3852630, 4618712, 5502170, 6516012

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A035596-A035606.

Column 5 of A035607, A266213. Row 5 of A113413, A119800, A122542.

[(4*n^4+20*n^2+6)*n/15: n in [0..30]]; // Vincenzo Librandi, Apr 23 2012

f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm

CoefficientList[Series[2*x*(1+x)^4/(1-x)^6,{x,0,33}],x] (* Vincenzo Librandi, Apr 23 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,20,102,360,1002},40] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=(4*n^5+20*n^3+6*n)/15 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = (4*n^4+20*n^2+6)*n/15. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^4/(1-x)^6. - Colin Barker, Mar 19 2012
a(n) = 2*A069038(n). - R. J. Mathar, Dec 10 2013
From Shel Kaphan, Mar 01 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=5.
a(n) = A001847(n) - A001846(n).
a(n) = A008413(n)*n/5. (End)

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 4, 1, 6, 18, 1, 8, 32, 88, 1, 10, 50, 170, 450, 1, 12, 72, 292, 912, 2364, 1, 14, 98, 462, 1666, 4942, 12642, 1, 16, 128, 688, 2816, 9424, 27008, 68464, 1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898

Offset: 0

R. J. Mathar, Apr 21 2021

			The full array starts
     1      2      2      2      2      2      2      2      2
     1      4      8     12     16     20     24     28     32
     1      6     18     38     66    102    146    198    258
     1      8     32     88    192    360    608    952   1408
     1     10     50    170    450   1002   1970   3530   5890
     1     12     72    292    912   2364   5336  10836  20256
     1     14     98    462   1666   4942  12642  28814  59906
     1     16    128    688   2816   9424  27008  68464 157184
     1     18    162    978   4482  16722  53154 148626 374274

J. Schroder, Generalized Schroder Numbers and the Rotation principle, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.

Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).

Main diagonal gives A050146(n+1).

A343599 := proc(n,k)
    local g,x,y ;
    g := (1+y)/(1-x-y-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:

T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

G.f.: (1+y)/(1-x-y-x*y).

T(n,k) = A008288(n,k) + A008288(n,k-1).

A019561 Coordination sequence for C_5 lattice.

Original entry on oeis.org

1, 50, 450, 1970, 5890, 14002, 28610, 52530, 89090, 142130, 216002, 315570, 446210, 613810, 824770, 1086002, 1404930, 1789490, 2248130, 2789810, 3424002, 4160690, 5010370, 5984050, 7093250, 8350002

Offset: 0

mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A103884 (row 5). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

Cf. A008413, A137513.

LinearRecurrence[{5,-10,10,-5,1},{1,50,450,1970,5890,14002},30] (* Harvey P. Dale, Nov 21 2021 *)

G.f.: (1+45*x+210*x^2+210*x^3+45*x^4+x^5)/(1-x)^5 = 1+2*x*(5+10*x+x^2)^2/(1-x)^5.
G.f. for sequence with interpolated zeros: cosh(10*arctanh(x)) = 1/2*( ((1 + x)/(1 - x))^5 + ((1 - x)/(1 + x))^5 ) = 1 + 50*x^2 + 450*x^4 + 1970*x^6 + .... - Peter Bala, Apr 09 2017
a(n) = A008413(2*n). - Seiichi Manyama, Jun 08 2018

A236967 Expansion of (1+3x)^2/(1-3x)^2.

Original entry on oeis.org

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

cp's OEIS Frontend

A113413 A Riordan array of coordination sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Sage

Formula

A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A008416 Coordination sequence for 8-dimensional cubic lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A019561 Coordination sequence for C_5 lattice.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Formula

Extensions

A236967 Expansion of (1+3x)^2/(1-3x)^2.