A050534 -id:A050534 - cp's OEIS frontend

A033486 a(n) = n(n + 1)(n + 2)*(n + 3)/2.

0, 12, 60, 180, 420, 840, 1512, 2520, 3960, 5940, 8580, 12012, 16380, 21840, 28560, 36720, 46512, 58140, 71820, 87780, 106260, 127512, 151800, 179400, 210600, 245700, 285012, 328860, 377580, 431520, 491040, 556512, 628320, 706860, 792540, 885780, 987012

Offset: 0

N. J. A. Sloane

a(n) is the area of an irregular quadrilateral with vertices at (1,1), (n+1, n+2), ((n+1)^2, (n+2)^2) and ((n+1)^3, (n+2)^3). - Art Baker, Dec 08 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..680 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A050534, A033488, A033487, A060008.

List([0..40],n->n*(n+1)*(n+2)*(n+3)/2); # Muniru A Asiru, Dec 08 2018

[n*(n+1)*(n+2)*(n+3)/2: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

[seq(12*binomial(n+3,4),n=0..32)]; # Zerinvary Lajos, Nov 24 2006

Table[n*(n + 1)*(n + 2)*(n + 3)/2, {n, 0, 50}] (* David Nacin, Mar 01 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,12,60,180,420},40] (* Harvey P. Dale, Feb 04 2015 *)

a(n)=n*(n+1)*(n+2)*(n+3)/2 \\ Charles R Greathouse IV, Oct 07 2015

[12*binomial(n+3,4) for n in range(40)] # G. C. Greubel, Dec 08 2018

a(n) = 6*A034827(n+3) = 12*A000332(n+3).
G.f.: 12*x/(1 - x)^5. - Colin Barker, Mar 01 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) with a(0) = 0, a(1) = 12, a(2) = 60, a(3) = 180, a(4) = 420. - Harvey P. Dale, Feb 04 2015
E.g.f.: (24*x + 36*x^2 + 12*x^3 + x^4)*exp(x)/2. - Franck Maminirina Ramaharo, Dec 08 2018
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*(3*log(2)-2)/9. (End)

A194480 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 1, 15, 10, 0, 0, 17, 45, 15, 0, 0, 6, 105, 105, 21, 0, 0, 0, 114, 410, 210, 28, 0, 0, 0, 39, 879, 1225, 378, 36, 0, 0, 0, 1, 909, 4284, 3066, 630, 45, 0, 0, 0, 0, 337, 8568, 15729, 6762, 990, 55, 0, 0, 0, 0, 15, 8733, 50526, 47565, 13560, 1485, 66, 0, 0, 0

Offset: 1

R. H. Hardin, Aug 26 2011

Table starts
...1....0......0........0.........0..........0...........0............0
...3....3......1........0.........0..........0...........0............0
...6...15.....17........6.........0..........0...........0............0
..10...45....105......114........39..........1...........0............0
..15..105....410......879.......909........337..........15............0
..21..210...1225.....4284......8568.......8733........3525..........285
..28..378...3066....15729.....50526......96478.......98473........43713
..36..630...6762....47565....221508.....668028.....1237434......1279905
..45..990..13560...124803....789453....3413828.....9821400.....17860056
..55.1485..25245...293733...2412333...14054915....57367112....159352995
..66.2145..44275...634293...6542316...49171641...268378248...1046727933
..78.3003..73931..1277133..16127397..151422970..1059987987...5488359255
..91.4095.118482..2426424..36762726..420674150..3661533037..24183257037
.105.5460.183365..4389567..78495417.1073422309.11341971885..92740471038
.120.7140.275380..7615062.158548572.2550004472.32090198922.317395080927
.136.9180.402900.12739902.305303544.5699074284.84099053568.987664967535

			Some solutions for n=4, k=4:
.....1........0........0........0........0........0........1........1
....1.0......1.0......0.1......0.1......1.0......1.1......0.1......0.1
...0.1.0....1.0.1....0.1.0....0.0.1....0.1.1....0.1.0....1.1.0....0.1.0
..0.0.0.1..0.0.0.1..1.0.1.0..0.1.1.0..0.0.0.1..1.0.0.0..0.0.0.0..0.0.1.0

R. H. Hardin, Table of n, a(n) for n = 1..290
Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 31-33.

Column 1 is A000217.

Column 2 is A050534(n+1).

Empirical: T(n,k) is a polynomial of degree 2k in n with lead coefficient 1/(2^k*k!) for k <= 5.
T(n,1) = (1/2)*n^2 + (1/2)*n
T(n,2) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (3/16)*n^4 + (1/48)*n^3 + (1/6)*n^2 - (1/12)*n
T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n
T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (7/384)*n^8 + (37/1920)*n^7 + (737/3840)*n^6 - (2347/3840)*n^5 + (101/192)*n^4 + (93/320)*n^3 - (7/10)*n^2 + (3/10)*n

A210569 a(n) = (n-3)(n-2)(n-1)n(n+1)/30.

Original entry on oeis.org

0, 0, 0, 0, 4, 24, 84, 224, 504, 1008, 1848, 3168, 5148, 8008, 12012, 17472, 24752, 34272, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 263120, 322920, 393120, 475020, 570024, 679644, 805504, 949344, 1113024, 1298528, 1507968, 1743588, 2007768, 2303028

Offset: 0

Bruno Berselli, Mar 23 2012

The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1):
. A000217(n)   for k=1,
. A007290(n+1) for k=2,
. A050534(n)   for k=3,
. a(n)         for k=4,
. A000910(n+1) for k=5.
Sum of reciprocals of a(n), for n>3: 5/16.
From a(2), convolution of oblong numbers (A002378) with themselves. - Bruno Berselli, Oct 24 2016

Bruno Berselli, Table of n, a(n) for n = 0..1000
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 3. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A000389, A000910, A002378, A007290, A050534, A177747.

First differences are in A033488.

Magma
```
[4*Binomial(n+1,5): n in [0..38]];
```

f:=n->(n^5-5*n^3+4*n)/30;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

LinearRecurrence[{6,-15,20,-15,6,-1}, {0,0,0,0,4,24}, 39]
CoefficientList[Series[4x^4/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Times@@@Partition[Range[-3,40],5,1]/30 (* Harvey P. Dale, Sep 19 2020 *)

makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38);

a(n)=(n-3)*(n-2)*(n-1)*n*(n+1)/30 \\ Charles R Greathouse IV, Oct 07 2015

[4*binomial(n+1,5) for n in (0..40)] # G. C. Greubel, May 23 2022

G.f.: 4*x^4/(1-x)^6.
a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1).
a(n) = 2*A177747(2*n-7), with A177747(-7) = A177747(-5) = A177747(-3) = A177747(-1) = 0.
(n-4)*a(n) = (n+1)*a(n-1).
E.g.f.: (1/30)*x^4*(5+x)*exp(x). - G. C. Greubel, May 23 2022
Sum_{n>=4} (-1)^n/a(n) = 20*log(2) - 655/48. - Amiram Eldar, Jun 02 2022

A240439 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 105, 114, 39, 3, 1, 15, 105, 420, 969, 1194, 654, 102, 3, 1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15, 1, 28, 378, 3150, 17415, 64776, 159528, 250233, 234609, 119259, 28395, 2613, 69, 1, 36, 630, 6930

Offset: 1

Heinrich Ludwig, Apr 05 2014

The triangle T(n, k) is irregularly shaped: 0 <= k <= A240114(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three of them form an equilateral triangle is given by A240114(n).

			The triangle begins:
  1,  1;
  1,  3,   3;
  1,  6,  15,   15,    3;
  1, 10,  45,  105,  114,    39,     3;
  1, 15, 105,  420,  969,  1194,   654,   102,    3;
  1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15;
There are T(5, 8) = 3 ways to place 8 points (x) on a triangular grid of side 5 under the conditions mentioned above:
          .                x                x
         x x              x .              . x
        x . x            x . .            . . x
       x . . x          x . . .          . . . x
      x . . . x        . x x x x        x x x x .

Heinrich Ludwig, Table of n, a(n) for n = 1..138

Cf. A240114, A240443, A084546,
column 2 is A000217,
column 3 is A050534,
column 4 is A240440,
column 5 is A240441,
column 6 is A240442.

A053526 Number of bipartite graphs with 3 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 16, 110, 435, 1295, 3220, 7056, 14070, 26070, 45540, 75790, 121121, 187005, 280280, 409360, 584460, 817836, 1124040, 1520190, 2026255, 2665355, 3464076, 4452800, 5666050, 7142850, 8927100, 11067966, 13620285

Offset: 0

N. J. A. Sloane, Jan 16 2000

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Column k=3 of A117279.

Cf. A000217 (1 edge), A050534 (2 edges).

List([0..40], n-> Binomial(n,4)*(n^2+3*n+4)/2) # G. C. Greubel, May 15 2019

[Binomial(n,4)*(n^2+3*n+4)/2: n in [0..40]]; // G. C. Greubel, May 15 2019

Table[Binomial[n,4]*(n^2+3*n+4)/2, {n,0,40}] (* G. C. Greubel, May 15 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,16,110,435},40] (* Harvey P. Dale, Nov 24 2022 *)

{a(n) = binomial(n,4)*(n^2+3*n+4)/2}; \\ G. C. Greubel, May 15 2019

[binomial(n,4)*(n^2+3*n+4)/2 for n in (0..40)] # G. C. Greubel, May 15 2019

a(n) = (n-3)*(n-2)*(n-1)*n*(n^2 + 3*n + 4)/48.
G.f.: x^4*(16-2*x+x^2)/(1-x)^7. - Colin Barker, May 08 2012
E.g.f.: x^4*(32 + 12*x + x^2)*exp(x)/48. - G. C. Greubel, May 15 2019

A060008 a(n) = 9binomial(n,4) = 3n(n-1)(n-2)(n-3)/8.

Original entry on oeis.org

0, 0, 0, 0, 9, 45, 135, 315, 630, 1134, 1890, 2970, 4455, 6435, 9009, 12285, 16380, 21420, 27540, 34884, 43605, 53865, 65835, 79695, 95634, 113850, 134550, 157950, 184275, 213759, 246645, 283185, 323640, 368280, 417384, 471240, 530145, 594405

Offset: 0

Henry Bottomley, Mar 16 2001

Number of permutations of n letters where exactly four change position.

			a(6) = 135 since there are 15 ways to choose the four points that move and 9 ways to move them and 15*9 = 135.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

For changing 0, 1, 2, 3, 4, 5, n-4, n elements see A000012, A000004, A000217 (offset), A007290, A060008, A060836, A000475, A000166. Also see A000332, A008290.

A diagonal of A008291.

9*Binomial[Range[0,40],4] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,0,9},40] (* Harvey P. Dale, Jun 09 2014 *)

a(n) = { 3*n*(n - 1)*(n - 2)*(n - 3)/8 } \\ Harry J. Smith, Jul 01 2009

Equals 3*A050534. - Zerinvary Lajos, Feb 12 2007
G.f.: 9*x^4/(1-x)^5. - Colin Barker, Jul 02 2012
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 4/27.
Sum_{n>=4} (-1)^n/a(n) = 32*log(2)/9 - 64/27. (End)

A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368

Offset: 0

Vladeta Jovovic, Jun 23 2007

			Triangle begins:
  1;
  1;
  1,  1;
  1,  3,  3;
  1,  6, 15,  16,   3;
  1, 10, 45, 110, 140, 60, 10;
  ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Andrew Howroyd, Table of n, a(n) for n = 0..1403 (rows 0..25)

Row sums give A047864,
Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
The unlabeled version is A297877.
Cf. A052296, A033995.

nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)

T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022

E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).

A144228 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph has at most one cycle.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 15, 1, 10, 45, 120, 210, 222, 1, 15, 105, 455, 1365, 2913, 3670, 1, 21, 210, 1330, 5985, 20139, 49294, 68820, 1, 28, 378, 3276, 20475, 97860, 362670, 976560, 1456875, 1, 36, 630, 7140, 58905, 376236, 1914276, 7663500, 22089870, 34506640

Offset: 0

Alois P. Heinz, Sep 15 2008

			T(4,4) = 15, because there are 15 simple graphs on 4 labeled nodes with 4 edges where each maximally connected subgraph has at most one cycle:
  1-2  1-2  1-2  1-2  1-2  1-2  1 2  1 2  1-2  1 2  1 2  1-2  1-2  1-2  1 2
  |/|  |X   |/   |\|   X|   \|  |/|   X|   /|  |\|  |X   |\   | |   X   |X|
  4 3  4 3  4-3  4 3  4 3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4-3  4 3
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  3,   1;
  1,  6, 15,  20,  15;
  1, 10, 45, 120, 210, 222;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

Columns k=0-3 give: A000012, A000217, A050534, A093566.
Main diagonal gives A137916.
Row sums give: A133686.
T(2n,n) gives A369828.
Cf. A000272, A057500, A007318, A000142, A129271.

cy:= proc(n) option remember; local t; binomial(n-1, 2) *add((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n,k) option remember; local j; if k=0 then 1 elif k<0 or n

t[, 0] = 1; t[n, k_] /; (k<0 || nJean-François Alcover, Jan 15 2014, after Maple *)

T(n,0) = 1, T(n,k) = 0 if k<0 or nA000272(j+1) T(n-j-1,k-j) + A057500(j+1) T(n-j-1,k-j-1)).

E.g.f.: exp(B(x,y)), where B(x,y) = Sum(Sum(A062734(n,k)*y^k*x^n/n!, k=0..n), n=1..infinity) = -1/2*log(1+LambertW(-x*y))+1/2*LambertW(-x*y) -1/4*LambertW(-x*y)^2-1/y *(LambertW(-x*y)+1/2 *LambertW(-x*y)^2). - Vladeta Jovovic, Sep 16 2008

A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3(x + 1)^n - 2(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 60, 45, 12, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 168, 210, 168, 84, 16, 1, 1, 18, 108, 252, 378, 378, 252, 108, 18, 1, 1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,   1;
  1, 10,  30,  30,  10,   1;
  1, 12,  45,  60,  45,  12,   1;
  1, 14,  63, 105, 105,  63,  14,   1;
  1, 16,  84, 168, 210, 168,  84,  16,  1;
  1, 18, 108, 252, 378, 378, 252, 108,  18,  1;
  1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A095151, A132046, A168643, A168644, A168646.

Columns (essentially): A005843 (k=1), A045943 (k=2), A027480 (k=3), A050534 (k=4), A253942 (k=5), A253943 (k=6), A253944 (k=7).

function f(n,k)
   if n le 2 then return 1;
   elif k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2;
   else return 3;
   end if;
end function;
A168641:= func< n,k | Binomial(n,k)*f(n,k) >;
[A168641(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 24 2025

p[x_, n_]:= If[n==0, 1, If[n==1, 1+x, 3*(1+x)^n -(1+x^n) -(1+n*x +n*x^(n-1) + x^n)]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{b=Boole}, If[k<=n/2, b[k==0] +2*b[k==1] +3*b[2<=k<=n/2], f[n, n-k]]];
A168641[n_, k_]:= Binomial[n,k]*If[n<3,1,f[n,k]];
Table[A168641[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2025 *)

T(n,k) := ratcoef(if n <= 2 then (1 + x)^n else 3*(x + 1)^n - (x^n + 1) - (x^n + n*x^(n - 1) + n*x + 1), x, k);
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

def f(n,k):
    if (k<=n/2): return int(k==0) + 2*int(k==1) + 3*int(1A168641(n,k):
    if (n<3): return binomial(n,k)
    else: return binomial(n,k)*f(n,k)
print(flatten([[A168641(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 24 2025

From G. C. Greubel, Mar 24 2025: (Start)
T(n, k) = 3*binomial(n, k), for n >= 4 and 2 <= k <= n-2, otherwise T(n, 0) = T(n, n) = 1, T(n, 1) = T(n, n-1) = 2*A065475(n-1).
T(n, n-k) = T(n, k).
T(n, 1) = A005843(n) - [n=1] - 2*[n=2].
Columns: T(n, k) = 3*binomial(n,k) - 2*[n=k] - (k+1)*[n=k+1], k >= 2.
Sum_{k=0..n} T(n, k) = 2*A095151(n-1) - 2*[n=0] - 2*[n=1].
Sum_{k=0..n} (-1)^k*T(n, k) = (1+(-1)^n)*(n-2) + 5*[n=0]. (End)

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A194485 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no four points are in the same row or diagonal.

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 1, 15, 10, 0, 0, 20, 45, 15, 0, 0, 15, 120, 105, 21, 0, 0, 6, 207, 455, 210, 28, 0, 0, 1, 234, 1347, 1330, 378, 36, 0, 0, 0, 165, 2817, 5922, 3276, 630, 45, 0, 0, 0, 63, 4135, 19362, 20307, 7140, 990, 55, 0, 0, 0, 9, 4080, 47010, 94584, 58527, 14190, 1485

Offset: 1

R. H. Hardin, Aug 26 2011

Table starts
...1....0......0.......0.........0..........0...........0............0
...3....3......1.......0.........0..........0...........0............0
...6...15.....20......15.........6..........1...........0............0
..10...45....120.....207.......234........165..........63............9
..15..105....455....1347......2817.......4135........4080.........2463
..21..210...1330....5922.....19362......47010.......83745.......105663
..28..378...3276...20307.....94584.....337860......927471......1931571
..36..630...7140...58527....365904....1790472.....6924357.....21123489
..45..990..14190..148239...1193283....7622340....39196161....162957252
..55.1485..26235..339669...3413619...27489825...180512640....974497260
..66.2145..45760..718344...8800704...87018360...708150465...4794685500
..78.3003..76076.1422564..20845968..247874770..2442836682..20207649891
..91.4095.121485.2666664..46017972..647091588..7582054194..75074999142
.105.5460.187460.4771221..95710797.1569661600.21540941994.251128663929
.120.7140.280840.8201466.189154056.3576049620.56763356130.768641935191

			Some solutions for n=5, k=4:
......0..........0..........0..........1..........0..........0..........0
.....0.0........0.1........0.1........1.0........0.0........0.1........0.0
....0.0.0......0.0.1......1.0.0......0.0.0......0.0.0......0.0.0......1.1.0
...0.0.1.1....0.0.0.0....0.0.1.0....0.0.1.0....0.0.1.1....0.0.0.0....0.0.0.0
..0.1.0.0.1..0.1.1.0.0..0.1.0.0.0..0.1.0.0.0..0.0.1.1.0..0.1.1.0.1..1.0.1.0.0

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A000217.
Column 2 is A050534(n+1).
Column 3 is A093566(n+2).

Empirical:
T(n,1) = (1/2)*n^2 + (1/2)*n
T(n,2) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,3) = (1/48)*n^6 + (1/16)*n^5 - (1/16)*n^4 - (11/48)*n^3 + (1/24)*n^2 + (1/6)*n
T(n,4) = (1/384)*n^8 + (1/96)*n^7 - (1/64)*n^6 - (13/120)*n^5 + (19/128)*n^4 + (7/96)*n^3 - (13/96)*n^2 + (1/40)*n
T(n,5) = (1/3840)*n^10 + (1/768)*n^9 - (1/384)*n^8 - (59/1920)*n^7 + (281/3840)*n^6 + (149/3840)*n^5 - (5/24)*n^4 + (29/320)*n^3 + (11/80)*n^2 - (1/10)*n
T(n,6) = (1/46080)*n^12 + (1/7680)*n^11 - (1/3072)*n^10 - (137/23040)*n^9 + (871/46080)*n^8 + (3107/161280)*n^7 - (5573/46080)*n^6 + (1157/23040)*n^5 + (2627/11520)*n^4 - (1121/5760)*n^3 - (181/1440)*n^2 + (11/84)*n
T(n,7) = (1/645120)*n^14 + (1/92160)*n^13 - (1/30720)*n^12 - (79/92160)*n^11 + (101/30720)*n^10 + (757/129024)*n^9 - (3049/92160)*n^8 - (34099/645120)*n^7 + (6613/15360)*n^6 - (16859/23040)*n^5 + (1043/3840)*n^4 + (2759/5040)*n^3 - (753/1120)*n^2 + (13/56)*n
Empirical: general T(n,k,z) for fewer than z points in any row or diagonal is polynomial in n of degree 2k with lead coefficient 1/(2^k*k!) for small k.
T(n,1,2) = (1/2)*n^2 + (1/2)*n
T(n,1,3) = (1/2)*n^2 + (1/2)*n
T(n,1,4) = (1/2)*n^2 + (1/2)*n
T(n,2,2) = (1/8)*n^4 - (1/4)*n^3 - (1/8)*n^2 + (1/4)*n
T(n,2,3) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,2,4) = (1/8)*n^4 + (1/4)*n^3 - (1/8)*n^2 - (1/4)*n
T(n,3,3) = (1/48)*n^6 + (1/16)*n^5 - (3/16)*n^4 + (1/48)*n^3 + (1/6)*n^2 - (1/12)*n
T(n,3,4) = (1/48)*n^6 + (1/16)*n^5 - (1/16)*n^4 - (11/48)*n^3 + (1/24)*n^2 + (1/6)*n
T(n,4,3) = (1/384)*n^8 + (1/96)*n^7 - (5/64)*n^6 + (13/240)*n^5 + (27/128)*n^4 - (23/96)*n^3 - (13/96)*n^2 + (7/40)*n
T(n,4,4) = (1/384)*n^8 + (1/96)*n^7 - (1/64)*n^6 - (13/120)*n^5 + (19/128)*n^4 + (7/96)*n^3 - (13/96)*n^2 + (1/40)*n
T(n,5,3) = (1/3840)*n^10 + (1/768)*n^9 - (7/384)*n^8 + (37/1920)*n^7 + (737/3840)*n^6 - (2347/3840)*n^5 + (101/192)*n^4 + (93/320)*n^3 - (7/10)*n^2 + (3/10)*n
T(n,5,4) = (1/3840)*n^10 + (1/768)*n^9 - (1/384)*n^8 - (59/1920)*n^7 + (281/3840)*n^6 + (149/3840)*n^5 - (5/24)*n^4 + (29/320)*n^3 + (11/80)*n^2 - (1/10)*n
T(n,6,4) = (1/46080)*n^12 + (1/7680)*n^11 - (1/3072)*n^10 - (137/23040)*n^9 + (871/46080)*n^8 + (3107/161280)*n^7 - (5573/46080)*n^6 + (1157/23040)*n^5 + (2627/11520)*n^4 - (1121/5760)*n^3 - (181/1440)*n^2 + (11/84)*n
T(n,7,4) = (1/645120)*n^14 + (1/92160)*n^13 - (1/30720)*n^12 - (79/92160)*n^11 + (101/30720)*n^10 + (757/129024)*n^9 - (3049/92160)*n^8 - (34099/645120)*n^7 + (6613/15360)*n^6 - (16859/23040)*n^5 + (1043/3840)*n^4 + (2759/5040)*n^3 - (753/1120)*n^2 + (13/56)*n

cp's OEIS Frontend

A033486 a(n) = n*(n + 1)*(n + 2)*(n + 3)/2.

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A194480 T(n,k) = number of ways to arrange k indistinguishable points on an n X n X n triangular grid so that no three points are in the same row or diagonal.

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A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.

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A240439 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

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A053526 Number of bipartite graphs with 3 edges on nodes {1..n}.

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A060008 a(n) = 9*binomial(n,4) = 3n*(n-1)*(n-2)*(n-3)/8.

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A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

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A033486 a(n) = n(n + 1)(n + 2)*(n + 3)/2.

A210569 a(n) = (n-3)(n-2)(n-1)n(n+1)/30.

A060008 a(n) = 9binomial(n,4) = 3n(n-1)(n-2)(n-3)/8.

A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3(x + 1)^n - 2(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.