A266213 -id:A266213 - cp's OEIS frontend

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066

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).
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 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 (5,-10,10,-5,1).

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

[( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 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)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)

a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

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)

A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.

Original entry on oeis.org

720, 13824, 383000, 14804640, 764340024, 50913153536, 4256161751448, 436618291524000, 53955264479804600, 7908071556041000064, 1356709951589099693976, 269380212536429979520928, 61297096735652845698099000, 15847986814197933588682229760, 4620315237160994963528810238104

Offset: 3

Eric W. Weisstein, Feb 20 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 3..200 (terms 3..50 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Graph Join.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to graphs, Hamiltonian

Cf. A130130, A234628, A266213, A270047.

b[n_, k_] := If[k == 0, 0, Sum[j*Min[2, j] * Sum[ Binomial[n - j - k, kk - 1]*Binomial[k - 1, kk]*2^kk, {kk, 0, Min[k - 1, n - j - k + 1]}], {j, 1, n - k + 1}]];
Flatten[{{2, 24}, Table[Sum[2*k!*b[n, k]*(k!*b[n, k] + (k - 1)!*b[n, k - 1]), {k, 1, n}], {n, 3, 20}]}] (* Vaclav Kotesovec, Mar 08 2016, after Andrew Howroyd *)

B(n)=polcoef(1/(1 - x*y*(2/(1 - x) - 1)) + O(x*x^n), n)
a(n)={my(v=Vecrev(B(n))); 2*n^2*sum(k=1, n, my(t=v[1+k]*(k-1)!); t*(t + if(k>1, v[k]*(k-2)!)))} \\ Andrew Howroyd, Jan 10 2025

a(n) = Sum_ { k=1..n } 2*k!*b(n,k)*(k!*b(n,k)+(k-1)!*b(n,k-1)) where b(n,0)=0, b(n,k)=Sum_{ j=1..n-k+1 } j*A130130(j)*A266213(k-1,n-j-k+1) for k>0, n<>2. - Andrew Howroyd, Feb 14 2016

a(n) ~ c * n!^2, where c = A270047 = 42.12277421168156081166292550105956... . - Vaclav Kotesovec, Mar 08 2016

a(6)-a(7) from Eric W. Weisstein, Dec 16 2013
a(8)-a(10) from Eric W. Weisstein, Dec 24 2013
a(1)=2 and a(2)=24 prepended by Andrew Howroyd, Feb 14 2016
a(11)-a(16) from Andrew Howroyd, Feb 14 2016
a(1)-a(2) removed by Andrew Howroyd, Jan 10 2025

A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 2, 16, 38, 1, 1, 2, 24, 146, 192, 1, 1, 2, 32, 326, 1408, 1002, 1, 1, 2, 40, 578, 4672, 14002, 5336, 1, 1, 2, 48, 902, 11008, 69002, 142000, 28814, 1, 1, 2, 56, 1298, 21440, 216002, 1038984, 1459810, 157184, 1, 1, 2, 64, 1766, 36992, 525002, 4320608, 15856206, 15158272, 864146, 1

Offset: 0

Seiichi Manyama, Jul 24 2020

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     2,     2,      2,      2, ...
  1,    8,    16,    24,     32,     40, ...
  1,   38,   146,   326,    578,    902, ...
  1,  192,  1408,  4672,  11008,  21440, ...
  1, 1002, 14002, 69002, 216002, 525002, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Column k=0-3 give A000012, A123164, A103885, A333715.
Main diagonal gives A336522.
Cf. A122542, A266213, A336534.

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n + j - 1, n - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 24 2020 *)

T(n,k) = (1/k) * [x^n] ( (1 + x)/(1 - x) )^(k*n) for k > 0 and n > 0.
T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j-1,n-1).
T(n,k) = (1/k) * Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j-1,j) for k > 0 and n > 0.
T(n,k) = Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n-1,j-1) for n > 0.
T(n,k) = binomial(k*n-1, n-1)*hypergeom([-n, k*n], [1+(k-1)*n], -1) for k > 0. - Stefano Spezia, Aug 09 2025

A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 30, 0, 1, 8, 30, 84, 170, 0, 1, 10, 48, 170, 496, 1050, 0, 1, 12, 70, 296, 1050, 3140, 6846, 0, 1, 14, 96, 470, 1920, 6846, 20832, 46374, 0, 1, 16, 126, 700, 3210, 12936, 46374, 142932, 323154, 0, 1, 18, 160, 994, 5040, 22402, 89712, 323154, 1005856, 2301618, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    6,    16,    30,    48,     70,     96, ...
  0,   30,    84,   170,   296,    470,    700, ...
  0,  170,   496,  1050,  1920,   3210,   5040, ...
  0, 1050,  3140,  6846, 12936,  22402,  36492, ...
  0, 6846, 20832, 46374, 89712, 159390, 266800, ...

Columns k=0..1 give A000007, A366266.
Main diagonal gives A378378.
Cf. A266213, A378317.
Cf. A378323.

T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(3/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A366266.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+2) for n > 0.

A035706 Coordination sequence for 11-dimensional cubic lattice.

Original entry on oeis.org

1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898, 26572086, 58227906, 121023606, 240089586, 457018518, 838478850, 1488341910, 2564399090, 4300978550, 7039035586, 11265589302, 17664712562, 27182654422

Offset: 0

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A035748, A266213 (row 11).

G.f.: ((1+x)/(1-x))^11.

n*a(n) = 22*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Aug 18 2018

Recomputed by N. J. A. Sloane, Nov 25 1998

A035734 Coordination sequence for 39-dimensional cubic lattice.

Original entry on oeis.org

1, 78, 3042, 79118, 1544322, 24138894, 314835170, 3525405390, 34608828930, 302685166030, 2388631358178, 17185219312014, 113694451659906, 696708049377294, 3979111519381986, 21295193543579982, 107295791104411650

Offset: 0

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, signature (39, -741, 9139, -82251, 575757, -3262623, 15380937, -61523748, 211915132, -635745396, 1676056044, -3910797436, 8122425444, -15084504396, 25140840660, -37711260990, 51021117810, -62359143990, 68923264410, -68923264410, 62359143990, -51021117810, 37711260990, -25140840660, 15084504396, -8122425444, 3910797436, -1676056044, 635745396, -211915132, 61523748, -15380937, 3262623, -575757, 82251, -9139, 741, -39, 1).

Cf. A035776, A266213 (row 39).

CoefficientList[Series[((1+n)/(1-n))^39,{n,0,20}],n] (* Harvey P. Dale, Apr 09 2015 *)

O.g.f.: ((1+x)/(1-x))^39. [clarified by Harvey P. Dale, Apr 09 2015]

n*a(n) = 78*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Aug 18 2018

Recomputed by N. J. A. Sloane, Nov 25 1998

A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2r+k,n)/(2r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 12, 0, 1, 8, 24, 40, 40, 0, 1, 10, 40, 92, 144, 144, 0, 1, 12, 60, 176, 360, 544, 544, 0, 1, 14, 84, 300, 752, 1440, 2128, 2128, 0, 1, 16, 112, 472, 1400, 3200, 5872, 8544, 8544, 0, 1, 18, 144, 700, 2400, 6352, 13664, 24336, 35008, 35008, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   2,    4,    6,     8,    10,    12, ...
  0,   4,   12,   24,    40,    60,    84, ...
  0,  12,   40,   92,   176,   300,   472, ...
  0,  40,  144,  360,   752,  1400,  2400, ...
  0, 144,  544, 1440,  3200,  6352, 11616, ...
  0, 544, 2128, 5872, 13664, 28480, 54768, ...

Columns k=0..1 give A000007, A025227(n+1).
Main diagonal gives A333473.
Cf. A266213, A378318.

T(n, k, t=0, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(2/k) )^k for k > 0.
G.f. of column k: (B(x)/x)^k where B(x) is the g.f. of A025227.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+1). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+1) for n > 0.

A035708 Coordination sequence for 13-dimensional cubic lattice.

Original entry on oeis.org

1, 26, 338, 2938, 19266, 101946, 454610, 1761370, 6065410, 18892250, 53972178, 143027898, 354870594, 830764794, 1847023698, 3921503898, 7988589570, 15677993370, 29746958930, 54734043130, 97926519106, 170763634106, 290835675858

Offset: 0

Joan Serra-Sagrista (jserra(AT)ccd.uab.es)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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.
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A035750, A266213 (row 13).

CoefficientList[Series[((1+x)/(1-x))^13,{x,0,30}],x] (* or *) LinearRecurrence[ {13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,26,338,2938,19266,101946,454610,1761370,6065410,18892250,53972178,143027898,354870594,830764794},30] (* Harvey P. Dale, Nov 07 2017 *)

G.f.: ((1+x)/(1-x))^13.

n*a(n) = 26*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Aug 18 2018

Recomputed by N. J. A. Sloane, Nov 25 1998

A125190 Number of ascents in all Schroeder paths of length 2n.

Original entry on oeis.org

0, 1, 6, 32, 170, 912, 4942, 27008, 148626, 822560, 4573910, 25534368, 143027898, 803467056, 4524812190, 25537728000, 144411206178, 818017823808, 4640757865126, 26364054632480, 149959897539018, 853941394691792, 4867745532495086, 27773897706129792

Offset: 0

Emeric Deutsch, Dec 20 2006

nonn

A Schroeder path of length 2n is a lattice path in the first quadrant, from the origin to the point (2n, 0) and consisting of steps U = (1, 1), D = (1, -1) and H = (2, 0); an ascent in a Schroeder path is a maximal strings of U steps.

a(n) is the number of points at L1 distance n - 2 from any point in Z^n, for n >= 2. - Shel Kaphan, Mar 24 2023

			a(2) = 6 because the Schroeder paths of length 4 are HH, H(U)D, (U)DH, (U)D(U)D, (U)HD and (UU)DD, having a total of 6 ascents (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 0..400
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

Cf. A006319, A090981, A008288, A026002, A190666.

-2-diagonal of A266213 for n>=1.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*R*(1+z*R)/sqrt(1-6*z+z^2): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..25);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<3, [0,1,6][n+1], ((204*n^2-411*n+198)*a(n-1)
       +(-34*n^2+68*n+96)*a(n-2) +(3*n-12)*a(n-3))/(2*n*(17*n-26)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 20 2012

CoefficientList[Series[x*(1-x-Sqrt[1-6*x+x^2])/(2*x)*(1+x*(1-x-Sqrt[1-6*x+x^2])/(2*x))/Sqrt[1-6*x+x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
a[n_] := Sum[ Binomial[n+1, n-i-1]*Binomial[n+i, n], {i, 0, n-1}]; (* or *) a[n_] := Hypergeometric2F1[1-n, 1+n, 3, -1]*n*(n+1)/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 05 2013, after Vladimir Kruchinin *)

A125190 = lambda n : (n^2+n)*hypergeometric([1-n, n+1], [3], -1)/2
[round(A125190(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 17 2014

a(n) = Sum_{k=0..n} k * A090981(n, k).
G.f.: z*R*(1 + z*R)/sqrt(1 - 6*z + z^2), where R = 1 + z*R + z*R^2, i.e., R = (1 - z -sqrt(1 - 6*z + z^2))/(2*z).
D-finite Recurrence: 2*n*(17*n - 26)*a(n) = 3*(68*n^2 - 137*n + 66)*a(n-1) - 2*(17*n^2 - 34*n - 48)*a(n-2) + 3*(n - 4)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(-3/4)*(3 + 2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2012
a(n) = Sum_{i=0..n-1} binomial(n+1, n-i-1) * binomial(n+i, n). - Vladimir Kruchinin, Feb 05 2013
a(n) = (n*(n+1)/2)*hypergeometric([1-n, n+1], [3], -1). - Peter Luschny, Sep 17 2014
a(n) = A026002(n) - A190666(n-2) for n >= 2. - Shel Kaphan, Mar 24 2023
a(n) = ((n+1)/2) * A006319(n-1). - Vladimir Kruchinin, Apr 27 2024

cp's OEIS Frontend

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

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

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A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.

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A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

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A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*r+k,n)/(3*r+k) for k > 0.

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PARI

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A035706 Coordination sequence for 11-dimensional cubic lattice.

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A035734 Coordination sequence for 39-dimensional cubic lattice.

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Mathematica

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A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*r+k,n)/(2*r+k) for k > 0.

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A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.

A378317 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2r+k,n)/(2r+k) for k > 0.