A003983 -id:A003983 - cp's OEIS frontend

A154957 A symmetric (0,1)-triangle.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Paul Barry, Jan 18 2009

Parity of A003983. - Jeremy Gardiner, Mar 09 2014

			Triangle begins
  1;
  1, 1;
  1, 0, 1;
  1, 0, 0, 1;
  1, 0, 1, 0, 1;
  1, 0, 1, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1;

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

Cf. A003983, A004524 (row sums), A154958 (diagonal sums), A158856.

T[n_, k_]:= Sum[(Mod[j+1,2] - Mod[j,2]), {j,0,Min[k,n-k]}];
Table[T[n, k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2022 *)

def A154957(n,k): return sum( (j+1)%2 - j%2 for j in (0..min(k,n-k)) )
flatten([[A154957(n,k) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Mar 07 2022

T(n,k) = Sum_{j=0..n} [j<=k]*[j<=n-k]*(mod(j+1,2) - mod(j,2)).
T(2*n, n) - T(2*n, n+1) = (-1)^n.
T(2*n, n) = (n+1) mod 2.
Sum_{k=0..n} T(n, k) = A004524(n+3).
Sum_{k=0..floor(n/2)} T(n-k, k) = A154958(n) (diagonal sums).
From G. C. Greubel, Mar 07 2022: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n, k) = floor((n+4)/4).
T(2*n+1, n) = (1+(-1)^n)/2. (End)

A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

Original entry on oeis.org

1, 1, 8, 1, 1, 8, 27, 8, 1, 1, 8, 27, 64, 27, 8, 1, 1, 8, 27, 64, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 729

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.

T(n,k) = min(n,k)^3. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

			Triangle starts
  1;
  1, 8, 1;
  1, 8, 27, 8, 1;
  1, 8, 27, 64, 27, 8, 1;
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...8...8...8...8...8...
  1...8..27..27..27..27...
  1...8..27..64..64..64...
  1...8..27..64.125.125...
  1...8..27..64.125.216...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  1,8,1;
  1,8,27,8,1;
  1,8,27,64,27,8,1;
  1,8,27,64,125,64,27,8,1;
  1,8,27,64,125,216,125,64,27,8,1;
  . . .
Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)

Harvey P. Dale, Table of n, a(n) for n = 0..9999
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.

Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* Harvey P. Dale, May 29 2019 *)

O.g.f.: (1+qx)(1+4qx+q^2x^2)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 8q + q^2) + x^2(1 + 8q + 27q^2 + 8q^3 + q^4) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^3.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)

A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

Original entry on oeis.org

1, 1, 16, 1, 1, 16, 81, 16, 1, 1, 16, 81, 256, 81, 16, 1, 1, 16, 81, 256, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 1296, 625, 256, 81, 16, 1, 1, 16, 81, 256, 625, 1296, 2401, 4096, 2401, 1296, 625, 256, 81, 16

Offset: 0

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.
From - Boris Putievskiy, Jan 13 2013: (Start)
The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1).
Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)

			Triangle starts:
  1;
  1, 16, 1;
  1, 16, 81, 16, 1;
  1, 16, 81, 256, 81, 16, 1;
  ...
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1.. .1...
  1..16..16..16..16...16...
  1..16..81..81..81...81...
  1..16..81.256.256..256...
  1..16..81.256.625..625...
  1..16..81.256.625.1296...
  ...
(End)

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.

p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* Harvey P. Dale, Dec 08 2014 *)

O.g.f.: (1+qx)(1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^4(1-q^2x)) = 1 + x(1 + 16q + q^2) + x^2(1 + 16q + 81q^2 + 16q^3 + q^4) + ... . Cf. 4th row of A008292.
From Boris Putievskiy, Jan 13 2013: (Start)
T(n,k) = min(n,k)^4.
a(n) = (A004737(n))^4.
a(n) = (A124258(n))^2.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)

A138139 Triangle read by rows: row n contains n terms and each column lists the prime numbers A000040.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 5, 3, 2, 2, 3, 5, 5, 3, 2, 2, 3, 5, 7, 5, 3, 2, 2, 3, 5, 7, 7, 5, 3, 2, 2, 3, 5, 7, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11

Offset: 1

Omar E. Pol, Mar 09 2008, Mar 25 2008

			Triangle begins:
      2
     2,2
    2,3,2
   2,3,3,2
  2,3,5,3,2

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A003983.

A138139 := proc(n,k) return ithprime(min(k,n-k+1)): end: seq(seq(A138139(n,k), k=1..n), n=1..13); # Nathaniel Johnston, Apr 30 2011

A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, -1, 0, 1, -4, 12, 64, -172, -1348, 3456, 34240, -87084, 370640, -872336, -22639616, 52307088, -181323568, 399580288, 23627011200, -51305628400, -686160247552, 1545932859328, 68098264912128, -155370174372864, 6326621032802304, -13829529077133312, -1087288396552040448

Offset: 0

Stefano Spezia, Apr 19 2022

sign

			a(8) = -172:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).

Join[{1},Table[Det[Table[If[Min[i,j]

a(n) = matdet(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304

Offset: 0

Stefano Spezia, Apr 19 2022

nonn

			a(8) = 7844:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).

Join[{1},Table[Permanent[Table[If[Min[i,j]

a(n) = matpermanent(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353453(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A115216 "Correlation triangle" for 2^n.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 8, 10, 10, 8, 16, 20, 21, 20, 16, 32, 40, 42, 42, 40, 32, 64, 80, 84, 85, 84, 80, 64, 128, 160, 168, 170, 170, 168, 160, 128, 256, 320, 336, 340, 341, 340, 336, 320, 256, 512, 640, 672, 680, 682, 682, 680, 672, 640, 512, 1024, 1280, 1344, 1360, 1364

Offset: 0

Paul Barry, Jan 16 2006

Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217.
Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
When formated as a square array, this is the self-fusion matrix (as in Example and Mathematica sections) of the sequence (2^n); for interlacing zeros of associated characteristic polynomials, see A202868.  [Clark Kimberling, Dec 26 2011]

			Triangle begins
  1,
  2, 2,
  4, 5, 4,
  8, 10, 10, 8,
  16, 20, 21, 20, 16,
  32, 40, 42, 42, 40, 32,
  ...
Northwest corner of square matrix:
  1....2....4....8....16
  2....5....10...20...40
  4....10...21...42...85
  8....20...41...85...170
  16...40...84...170..341
  ..

Cf. A003983, A202678.

(* A115216 as a square matrix *)
s[k_] := 2^(k - 1);
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* -1+2^n *)
Table[m[n, n], {n, 1, 12}]  (* A002450 *)
(* Clark Kimberling, Dec 26 2011 *)

T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).

G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006

A261684 Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 10, 2, 3, 4, 5, 5, 4, 3, 2, 10, 0, 0, 11, 10, 3, 4, 5, 6, 5, 4, 3, 10, 11, 0

Offset: 0

N. J. A. Sloane, Sep 06 2015

See A087061 for definition. Note that 0+x = x and 9*x = x for all x.

			Lunar multiplication table begins:
0 0 0 0 0 0 ...
0 1 1 1 1 1 ...
0 1 2 2 2 2 ...
0 1 2 3 3 3 ...
0 1 2 3 4 4 ...
0 1 2 3 4 5 ...
....

N. J. A. Sloane, Table of n, a(n) for n = 0..20099
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087061 (addition).
See A087062 for a version that excludes the zero row and column.
Similar to but different from A003983.

# convert decimal to string:
rec := proc(n) local t0,t1,e,l; if n <= 0 then RETURN([[0],1]); fi; t0 := n mod 10; t1 := (n-t0)/10; e := [t0]; l := 1; while t1 <> 0 do t0 := t1 mod 10; t1 := (t1-t0)/10; l := l+1; e := [op(e),t0]; od; RETURN([e,l]); end;
# convert string to decimal:
cer := proc(ep) local i,e,l,t1; e := ep[1]; l := ep[2]; t1 := 0; if l <= 0 then RETURN(t1); fi; for i from 1 to l do t1 := t1+10^(i-1)*e[i]; od; RETURN(t1); end;
# lunar addition:
dadd := proc(m,n) local i,r1,r2,e1,e2,l1,l2,l,l3,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := max(l1,l2); l3 := min(l1,l2); t0 := array(1..l); for i from 1 to l3 do t0[i] := max(e1[i],e2[i]); od; if l>l3 then for i from l3+1 to l do if l1>l2 then t0[i] := e1[i]; else t0[i] := e2[i]; fi; od; fi; cer([t0,l]); end;
# lunar multiplication:
dmul := proc(m,n) local k,i,j,r1,r2,e1,e2,l1,l2,l,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := l1+l2-1; t0 := array(1..l); for i from 1 to l do t0[i] := 0; od; for i from 1 to l2 do for j from 1 to l1 do k := min(e2[i],e1[j]); t0[i+j-1] := max(t0[i+j-1],k); od; od; cer([t0,l]); end;
# to produce the b-file:
M:=199; c:=0; for n from 0 to M do for k from 0 to n do lprint(c,dmul(n-k,k)); c:=c+1; od: od:

A352967 Array read by antidiagonals: A(i, j) = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0, with i >= 0 and j >= 0.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Apr 21 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 0, 1, 0, 0, 0, 0, 0, ...
    0, 1, 0, 1, 2, 0, 0, 0, ...
    0, 0, 1, 0, 1, 2, 3, 0, ...
    0, 0, 2, 1, 0, 1, 2, 3, ...
    0, 0, 0, 2, 1, 0, 1, 2, ...
    0, 0, 0, 3, 2, 1, 0, 1, ...
    0, 0, 0, 0, 3, 2, 1, 0, ...
    ...

Cf. A003983, A049581, A051125, A307018 (antidiagonal half-sums), A353452, A353453.

Mathematica
```
A[i_,j_]:=If[Min[i, j]
				
```

A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.

Original entry on oeis.org

1, 1, 5, 72, 2309, 140400, 14495641, 2347782144, 562385930985, 190398813728000, 87889475202276461, 53726132414026874880, 42454821207656237294381, 42495322215073539046387712, 52954624815227996007075890625, 80932107560443542398970529579008, 149736953621087625813286348913927569

Offset: 0

Stefano Spezia, Apr 29 2023

nonn

M(n-1)/n is the inverse of the Cartan matrix for SU(n): the special unitary group of degree n.
The elements sum of the matrix M(n) is A002415(n+1).
The antidiagonal sum of the matrix M(n) is A005993(n-1).
The n-th row of A107985 gives the row or column sums of the matrix M(n+1).

			a(3) = 72:
           [3, 2, 1]
    M(3) = [2, 4, 2]
           [1, 2, 3]
a(5) = 140400:
           [5, 4, 3, 2, 1]
           [4, 8, 6, 4, 2]
    M(5) = [3, 6, 9, 6, 3]
           [2, 4, 6, 8, 4]
           [1, 2, 3, 4, 5]

E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Translations, Series 2, Vol. 6, 1957.

Chao Ju, Chern-Simons Theory, Ehrhart Polynomials, and Representation Theory, arXiv:2304.11830 [math-ph], 2023. See p. 14.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes
Wikipedia, Special unitary group

Cf. A000272, A000292 (trace), A002415, A003983, A003991, A005993, A106314 (antidiagonals), A107985.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
    Matrix(n, (i, j)-> min(i, j)*(n+1)-i*j))):
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 30 2023

M[i_, j_, n_]:=Min[i, j](n+1)-i j; Join[{1}, Table[Permanent[Table[M[i, j, n], {i, n}, {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n, n, i, j, min(i, j)*(n + 1) - i*j)); \\ Michel Marcus, Apr 30 2023

Conjecture: det(M(n)) = A000272(n+1).

The conjecture is true (see proof in Links). - Stefano Spezia, May 24 2023

cp's OEIS Frontend

A154957 A symmetric (0,1)-triangle.

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A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

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A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

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A138139 Triangle read by rows: row n contains n terms and each column lists the prime numbers A000040.

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Maple

A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A115216 "Correlation triangle" for 2^n.

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A261684 Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals.

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A362679 a(n) is the permanent of the n X n symmetric matrix M(n) defined by M[i, j, n] = min(i, j)(n + 1) - ij.