A198063 -id:A198063 - cp's OEIS frontend

A073254 Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.

0, 1, 1, 4, 3, 4, 9, 7, 7, 9, 16, 13, 12, 13, 16, 25, 21, 19, 19, 21, 25, 36, 31, 28, 27, 28, 31, 36, 49, 43, 39, 37, 37, 39, 43, 49, 64, 57, 52, 49, 48, 49, 52, 57, 64, 81, 73, 67, 63, 61, 61, 63, 67, 73, 81, 100, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 121, 111, 103, 97

Offset: 0

Michael Somos, Jul 23 2002

Norm of elements in planar hexagonal lattice A_2.

Only numbers which appear in A003136 (Loeschian numbers) can appear in this array. - Peter Luschny, Nov 10 2021

			Triangle T(n, k) starts:
[0]               0
[1]              1, 1
[2]            4, 3, 4
[3]           9, 7, 7, 9
[4]       16, 13, 12, 13, 16
[5]     25, 21, 19, 19, 21, 25
[6]   36, 31, 28, 27, 28, 31, 36
[7] 49, 43, 39, 37, 37, 39, 43, 49

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (triangular rows 0 <= n <= 150, flattened)
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Index entries for sequences related to A2 = hexagonal = triangular lattice

A033994 gives antidiagonal sums.
Cf. A003136, A000290, A033428, A003215, A033994, A194595.
Cf. A004016, A057427, A003056, A132111.
Cf. A198063 (m=3), A198064 (m=4), A198065 (m=5).

# Using the triangle formula:
A073254 := (n,k) -> k^2 - k*n + n^2: # Peter Luschny, Oct 26 2011

(* Using the array formula: *)
A[n_, k_] := n^2 + n k + k^2;
Table[A[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

PARI
```
{A(n, k) = n^2 + n*k + k^2}
```

From Peter Luschny, Oct 26 2011: (Start)
Let m = 2, for the cases m = 3, 4, and 5 see the cross-references.
T(n,k) = k^2 - k*n + n^2 = A(n-k,k).
T(n,k) = Sum_{j=0..m} Sum_{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j) for m = 2.
T(n,0) = T(n,n) = n^m = n^2 = A000290(n).
T(2n,n) = (m+1)*n^m = 3*n^2 = A033428(n).
T(2n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^3 - n^3 = A003215(n).
Sum_{k=0..n} T(n,k) = (5*n^3 + 6*n^2 + n)/6 = A033994(n).
T(n+1, k+1)*binomial(n, k)^3/(k+1)^2 = A194595(n,k). (End)

Edited by Peter Luschny, Nov 10 2021

A197653 Triangle by rows T(n,k), showing the number of meanders with length (n+1)4 and containing (k+1)4 Ls and (n-k)*4 Rs, where Ls and Rs denote arcs of equal length and a central angle of 90 degrees which are positively or negatively oriented.

Original entry on oeis.org

1, 4, 1, 15, 30, 1, 40, 324, 120, 1, 85, 2080, 3120, 340, 1, 156, 9375, 40000, 18750, 780, 1, 259, 32886, 328125, 437500, 82215, 1554, 1, 400, 96040, 1959216, 6002500, 3265360, 288120, 2800, 1

Offset: 0

Susanne Wienand, Oct 17 2011

Definition of a meander:
A binary curve C is a triple (m, S, dir) such that
(a) S is a list with values in {L,R} which starts with an L,
(b) dir is a list of m different values, each value of S being allocated a value of dir,
(c) consecutive Ls increment the index of dir,
(d) consecutive Rs decrement the index of dir,
(e) the integer m > 0 divides the length of S and
(f) C is a meander if each value of dir occurs length(S)/m times.
For this sequence, m = 4.
The values in the triangle are proved by brute force for 0 <= n <= 8. The formulas are not yet proved in general.
The number triangle can be calculated recursively by the number triangles A007318, A103371 and A194595. The first column of the triangle seems to be A053698. The diagonal right hand is A000012. The diagonal with k = n-1 seems to be A027445. Row sums are in A198256.
The conjectured formulas are confirmed by dynamic programming for 0 <= n <= 43. - Susanne Wienand, Jun 29 2015

			For n = 4 and k = 2, T(4,4,2) = 3120.
Recursive example:
T(1,4,0) = 1,
T(1,4,1) = 4,
T(1,4,2) = 6,
T(1,4,3) = 4,
T(1,4,4) = 1,
T(3,4,0) = 21,
T(3,4,1) = 304,
T(3,4,2) = 456,
T(3,4,3) = 84,
T(3,4,1) = 1,
T(4,4,2) = 6^4 + 6*304 = 3120.
Example for closed formula:
T(4,2) = 6^4 + 6^3 * 4 + 6^2 * 4^2 + 6 * 4^3 = 3120.
Some examples of list S and allocated values of dir if n = 4 and k = 2:
Length(S) = (4+1)*4 = 20 and S contains (2+1)*4 = 12 Ls.
  S: L,L,L,L,L,L,L,L,L,L,L,L,R,R,R,R,R,R,R,R
dir: 1,2,3,0,1,2,3,0,1,2,3,0,0,3,2,1,0,3,2,1
  S: L,L,L,R,L,L,R,L,L,R,R,L,L,L,R,L,L,R,R,R
dir: 1,2,3,3,3,0,0,0,1,1,0,0,1,2,2,2,3,3,2,1
  S: L,R,L,L,L,L,R,R,R,L,L,R,R,L,L,L,R,L,L,R
dir: 1,1,1,2,3,0,0,3,2,2,3,3,2,2,3,0,0,0,1,1
Each value of dir occurs 20/4 = 5 times.
Triangle begins:
   1;
   4,    1;
  15,   30,    1;
  40,  324,  120,   1;
  85, 2080, 3120, 340, 1;
  ...

Susanne Wienand, Table of n, a(n) for n = 0..989
Peter Luschny, Meanders and walks on the circle.

Cf. A000012, A007318, A027445, A053698, A103371, A194595, A197654, A197655, A198063, A198256.

A197653 := (n,k) -> binomial(n,k)^4*(n+1)*(n^2-2*n*k+1+2*k+2*k^2)/((1+k)^3);
seq(print(seq(A197653(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 19 2011

T[n_, k_] := Binomial[n, k]^4 (n+1)(n^2 - 2n*k + 1 + 2k + 2k^2)/((1+k)^3);
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Peter Luschny *)

A197653(n,k) = {if(n==1+2*k,4,(1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n))*binomial(n,k)^4} \\ Peter Luschny, Nov 24 2011

def S(N,n,k) : return binomial(n,k)^(N+1)*sum(sum((-1)^(N-j+i)*binomial(N-i,j)*((n+1)/(k+1))^j for i in (0..N) for j in (0..N)))
def A197653(n,k) : return S(3,n,k)
for n in (0..5) : print([A197653(n,k) for k in (0..n)])  ## Peter Luschny, Oct 24 2011

Recursive formula (conjectured):
T(n,k)   = T(4,n,k)
T(4,n,k) = T(1,n,k)^4 + T(1,n,k)*T(3,n,n-1-k), 0 <= k < n
T(4,n,n) = 1                                        k = n
T(3,n,k) = T(1,n,k)^3 + T(1,n,k)*T(2,n,n-1-k), 0 <= k < n
T(3,n,n) = 1                                        k = n
T(2,n,k) = T(1,n,k)^2 + T(1,n,k)*T(1,n,n-1-k), 0 <= k < n
T(2,n,n) = 1                                        k = n
T(3,n,k) = A194595
T(2,n,k) = A103371
T(1,n,k) = A007318 (Pascal's Triangle)
closed formula (conjectured): T(n,n) = 1,                         k = n
                T(n,k) = A + B + C + D,             k < n
                     A = (C(n,k))^4
                     B = (C(n,k))^3 * C(n,n-1-k)
                     C = (C(n,k))^2 *(C(n,n-1-k))^2
                     D =  C(n,k)    *(C(n,n-1-k))^3
[Susanne Wienand]
Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,3). - Peter Luschny, Oct 20 2011
T(n,k) = A198063(n+1,k+1)*C(n,k)^4/(k+1)^3. - Peter Luschny, Oct 29 2011
T(n,k) = h(n,k)*binomial(n,k)^4, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n) if 1+2*k-n <> 0, otherwise h(n,k) = 4. - Peter Luschny, Nov 24 2011

A198064 Triangle read by rows (n >= 0, 0 <= k <= n, m = 4); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

0, 1, 1, 16, 5, 16, 81, 31, 31, 81, 256, 121, 80, 121, 256, 625, 341, 211, 211, 341, 625, 1296, 781, 496, 405, 496, 781, 1296, 2401, 1555, 1031, 781, 781, 1031, 1555, 2401, 4096, 2801, 1936, 1441, 1280, 1441, 1936, 2801, 4096, 6561, 4681, 3355, 2511, 2101

Offset: 0

Peter Luschny, Oct 26 2011

			[0]                      0
[1]                     1, 1
[2]                  16, 5, 16
[3]                81, 31, 31, 81
[4]            256, 121, 80, 121, 256
[5]         625, 341, 211, 211, 341, 625
[6]     1296, 781, 496, 405, 496, 781, 1296
[7] 2401, 1555, 1031, 781, 781, 1031, 1555, 2401

Cf. A057427, A003056, A073254, A198063, A198065.

A198064 := (n,k) -> k^4-2*k^3*n+4*k^2*n^2-3*k*n^3+n^4:

T(n,k) = k^4-2*k^3*n+4*k^2*n^2-3*k*n^3+n^4.
T(n,0) = T(n,n) = n^m = n^4 = A000583(n).
T(2n,n) = (m+1)n^m = 5n^4.
T(2n+1,n+1) = (n+1)^(m+1)-n^(m+1) = (n+1)^5-n^5 = A022521(n).
Sum{k=0..n} T(n,k) = (16n^5+30n^4+15n^3-n)/30.
T(n+1,k+1)C(n,k)^5/(k+1)^4 = A197654(n,k).

A198065 Triangle read by rows (n >= 0, 0 <= k <= n, m = 5); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

0, 1, 1, 32, 6, 32, 243, 63, 63, 243, 1024, 364, 192, 364, 1024, 3125, 1365, 665, 665, 1365, 3125, 7776, 3906, 2016, 1458, 2016, 3906, 7776, 16807, 9331, 5187, 3367, 3367, 5187, 9331, 16807, 32768, 19608, 11648, 7448, 6144, 7448, 11648, 19608, 32768, 59049

Offset: 0

Peter Luschny, Oct 26 2011

			[0]                        0
[1]                       1, 1
[2]                    32, 6, 32
[3]                 243, 63, 63, 243
[4]            1024, 364, 192, 364, 1024
[5]         3125, 1365, 665, 665, 1365, 3125
[6]     7776, 3906, 2016, 1458, 2016, 3906, 7776
[7] 16807, 9331, 5187, 3367, 3367, 5187, 9331, 16807

Cf. A057427, A003056, A073254, A198063, A198064.

Cf. A000584, A022522, A197655.

&cat[[n*(k^2-k*n+n^2)*(3*k^2-3*k*n+n^2): k in [0..n]]: n in [0..9]];  // Bruno Berselli, Nov 02 2011

A198065 := (n,k) -> 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5:

T(n,k) = 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5.
T(n,0) = T(n,n) = n^m = n^5 = A000584(n).
T(2n,n) = (m+1)n^m = 6n^5.
T(2n+1,n+1) = (n+1)^(m+1)-n^(m+1) = (n+1)^6-n^6 = A022522(n).
Sum{k=0..n} T(n,k) = (13n^6+30n^5+20n^4-3n^2)/30.
T(n+1,k+1)C(n,k)^6/(k+1)^5 = A197655(n,k).

A321500 Triangular table T(n,k) = (n+k)*(n^2+k^2), n >= k >= 0; read by rows n = 0, 1, 2, ...

Original entry on oeis.org

0, 1, 4, 8, 15, 32, 27, 40, 65, 108, 64, 85, 120, 175, 256, 125, 156, 203, 272, 369, 500, 216, 259, 320, 405, 520, 671, 864, 343, 400, 477, 580, 715, 888, 1105, 1372, 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048

Offset: 0

M. F. Hasler, Nov 22 2018

			The table starts:
n | T(n,k), k = 0..n:
0 |   0;
1 |   1,   4;
2 |   8,  15,  32;
3 |  27,  40,  65, 108;
4 |  64,  85, 120, 175, 256;
5 | 125, 156, 203, 272, 369,  500;
6 | 216, 259, 320, 405, 520,  671,  864;
7 | 343, 400, 477, 580, 715,  888, 1105, 1372;
8 | 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.

M. F. Hasler, Rows n = 0..141 of triangle, flattened

Cf. A000578 (column 0: the cubes), A033430 (diagonal: 4*n^3), A053698 (column 1).

Cf. A198063 (read as A(n,k)=(n+k)*(n^2+k^2)).

[[(n+k)*(n^2+k^2): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 23 2018

t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

A321500(n, k)=(n+k)*(n^2+k^2)
A321500_row(n)=vector(n+1, k, (n+k--)*(n^2+k^2))
A321500_list(N=11)=concat(apply(A321500_row, [0..N]))

[[(n+k)*(n^2+k^2) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 23 2018

Sum_{k=0..n} T(n,k) = 5*n^2*(n+1)*(5*n+1)/12 = 5*A117066(n). - G. C. Greubel, Nov 23 2018

A348897 Numbers of the form (x + y)*(x^2 + y^2).

Original entry on oeis.org

0, 1, 4, 8, 15, 27, 32, 40, 64, 65, 85, 108, 120, 125, 156, 175, 203, 216, 256, 259, 272, 320, 343, 369, 400, 405, 477, 500, 512, 520, 580, 585, 671, 680, 715, 729, 803, 820, 864, 888, 935, 960, 1000, 1080, 1105, 1111, 1157, 1248, 1261, 1331, 1372, 1400, 1417

Offset: 1

Peter Luschny, Nov 10 2021

nonn

Also numbers of the form (x - i*y)*(x + i*y)*(x + y).
Loeschian numbers of this form are A349200.
A349201 and A349202 are subsequences of this sequence.
Numbers of the form 1 + n + n^2 + n^3 (A053698) are a subsequence.
Numbers of the form n^3 + n^4 + n^5 + n^6 are a subsequence.
Numbers of the form 1 + n^2 + n^4 + n^6 (A059830) are a subsequence. - Bernard Schott, Nov 11 2021

			1010101 is in this sequence because 1010101 = (100 + 1)*(100^2 + 1^2).

Peter Luschny, Table of n, a(n) for n = 1..1200

Cf. A198063, A321500, A003136, A053698, A059830, A349200, A349201, A349202.

# Returns the terms less than or equal to b^3.
function A348897List(b)
    b3 = b^3; R = [0]
    for n in 1:b
        for k in 0:n
            a = (n + k) * (n^2 + k^2)
            a > b3 && break
            push!(R, a)
    end end
unique!(sort!(R)) end
A348897List(12) |> println

# Returns the terms less than or equal to b^3.
A348897List := proc(b) local n, k, a, b3, R;
b3 := abs(b^3); R := {};
for n from 0 to b do for k from 0 to n do
    a := (n + k)*(n^2 + k^2);
    if a > b3 then break fi;
    R := R union {a};
od od; sort(R) end:
A348897List(12);

max = 2000;
xmax = max^(1/3) // Ceiling;
Table[(x + y) (x^2 + y^2), {x, 0, xmax}, {y, x, xmax}] // Flatten // Union // Select[#, # <= max&]& (* Jean-François Alcover, Oct 23 2023 *)

A321490 Triangular table T[n,k] = (n+k)(n^2+k^2), 1 <= k <= n = 1, 2, 3, ...; read by rows.

Original entry on oeis.org

4, 15, 32, 40, 65, 108, 85, 120, 175, 256, 156, 203, 272, 369, 500, 259, 320, 405, 520, 671, 864, 400, 477, 580, 715, 888, 1105, 1372, 585, 680, 803, 960, 1157, 1400, 1695, 2048, 820, 935, 1080, 1261, 1484, 1755, 2080, 2465, 2916, 1111, 1248, 1417, 1624, 1875, 2176, 2533, 2952, 3439, 4000, 1464, 1625, 1820, 2055, 2336

Offset: 1

M. F. Hasler, Nov 22 2018

			The table starts:
Row 1:    4;
Row 2:   15,  32;
Row 3:   40,  65, 108;
Row 4:   85, 120, 175, 256;
Row 5:  156, 203, 272, 369,  500;
Row 6:  259, 320, 405, 520,  671,  864;
Row 7:  400, 477, 580, 715,  888, 1105, 1372;
Row 8:  585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.

Cf. A321491 (numbers of the form T(n,k) with n > k > 0).
Cf. A321492 (numbers which can be written at least twice in this form).
Cf. A033430 (diagonal), A053698 (column 1).
Cf. A198063 (read as a square array equals T(n,k) for all n, k >= 0).
Cf. A321500 (variant of this table with additional row 0 and column 0).

t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

A321490(n,k)=(n+k)*(n^2+k^2)
A321490_row(n)=vector(n,k,(n+k)*(n^2+k^2))
A321490_list(N=12)=concat(apply(A321490_row,[1..N]))

Diagonal: T(n,n) = 4*n^3 = A033430(n).

Column 1: T(n,1) = (n + 1)(n^2 + 1) = A053698(n) = (n^4-1)/(n-1) for n > 1.

A198062 Array read by antidiagonals, m>=0, n>=0, k>=0, A(m, n, k) = sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 4, 2, 1, 0, 1, 1, 8, 3, 2, 1, 0, 1, 1, 16, 4, 4, 3, 1, 0, 1, 1, 32, 5, 8, 9, 3, 1, 0, 1, 1, 64, 6, 16, 27, 7, 3, 1, 0, 1, 1, 128, 7, 32, 81, 15, 7, 3, 1, 0, 1, 1, 256, 8, 64, 243, 31, 15, 9, 4, 1, 0, 1, 1, 512, 9

Offset: 0

Peter Luschny, Nov 02 2011

			   [0] [1] [2]  [3] [4]  [5]  [6]  [7]  [8]  [9]
-------------------------------------------------
[0]  1   1   1    1   1    1    1    1    1    1    A000012
[1]  0   1   1    2   2    2    3    3    3    3    A003056
[2]  0   1   1    4   3    4    9    7    7    9    A073254
[3]  0   1   1    8   4    8   27   15   15   27    A198063
[4]  0   1   1   16   5   16   81   31   31   81    A198064
[5]  0   1   1   32   6   32  243   63   63  243    A198065

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A198060, A198061.

A198062_RowAsTriangle := proc(m) local pow; pow :=(a,b)->`if`(a=0 and b=0,1,a^b): proc(n, k) local i, j; add(add((-1)^(j + i)*binomial(i, j)*pow(n, j)* pow(k, m-j), i=0..m),j=0..m) end: end:
for m from 0 to 2 do seq(print(seq(A198062_RowAsTriangle(m)(n,k),k=0..n)),n=0..5) od;

max = 9; RowAsTriangle[m_][n_, k_] := Module[{pow}, pow[a_, b_] := If[a == 0 && b == 0, 1, a^b]; Module[{i, j}, Sum[Sum[(-1)^(j+i)*Binomial[i, j]*pow[n, j]*pow[k, m-j], {i, 0, m}], {j, 0, m}]]]; t = Flatten /@ Table[RowAsTriangle[m][n, k], {m, 0, max}, {n, 0, max}, {k, 0, n}]; Table[t[[n-k+1, k+1]], {n, 0, max}, {k, 0, n }] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

A007318(n,k) = A(0,n+1,k+1)*C(n,k)^1/(k+1)^0,
A103371(n,k) = A(1,n+1,k+1)*C(n,k)^2/(k+1)^1,
A194595(n,k) = A(2,n+1,k+1)*C(n,k)^3/(k+1)^2,
A197653(n,k) = A(3,n+1,k+1)*C(n,k)^4/(k+1)^3,
A197654(n,k) = A(4,n+1,k+1)*C(n,k)^5/(k+1)^4,
A197655(n,k) = A(5,n+1,k+1)*C(n,k)^6/(k+1)^5.

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A073254 Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.

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A197653 Triangle by rows T(n,k), showing the number of meanders with length (n+1)*4 and containing (k+1)*4 Ls and (n-k)*4 Rs, where Ls and Rs denote arcs of equal length and a central angle of 90 degrees which are positively or negatively oriented.

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A198064 Triangle read by rows (n >= 0, 0 <= k <= n, m = 4); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

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A198065 Triangle read by rows (n >= 0, 0 <= k <= n, m = 5); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

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Magma

Maple

Formula

A321500 Triangular table T(n,k) = (n+k)*(n^2+k^2), n >= k >= 0; read by rows n = 0, 1, 2, ...

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Magma

Mathematica

PARI

Sage

Formula

A348897 Numbers of the form (x + y)*(x^2 + y^2).

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Julia

Maple

Mathematica

A321490 Triangular table T[n,k] = (n+k)(n^2+k^2), 1 <= k <= n = 1, 2, 3, ...; read by rows.

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A197653 Triangle by rows T(n,k), showing the number of meanders with length (n+1)4 and containing (k+1)4 Ls and (n-k)*4 Rs, where Ls and Rs denote arcs of equal length and a central angle of 90 degrees which are positively or negatively oriented.

A198064 Triangle read by rows (n >= 0, 0 <= k <= n, m = 4); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

A198065 Triangle read by rows (n >= 0, 0 <= k <= n, m = 5); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).

A198062 Array read by antidiagonals, m>=0, n>=0, k>=0, A(m, n, k) = sum{j=0..m} sum{i=0..m} (-1)^(j+i)C(i,j)n^j*k^(m-j).