A023999 -id:A023999 - cp's OEIS frontend

A078475 Determinant of rank n matrix of 1..n^2 filled successively back and forth along antidiagonals.

1, -2, 15, -594, -5187, 23244, 122475, -279292, -1157143, 1850930, 6642839, -8529278, -27810555, 30741424, 93575187, -92784984, -268191855, 244875462, 679807583, -581798410, -1563707379, 1270245108, 3324627195, -2587197204, -6623079687, 4972012474, 12491212135

Offset: 1

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

The matrix is formed by writing numbers 1 .. n^2 in zig-zag pattern as shown in examples below. Every other antidiagonal reads backwards from A069480.
Whereas each antidiagonal of A069480 begins with one more than a triangular number and ends with the next triangular number, here every other antidiagonal begins with one more than a triangular number and the next antidiagonal begins with a triangular number.
The trace of the matrix is the sequence A006003 (proved). - Stefano Spezia, Aug 07 2018
The matrix is defined by A[i,j] = (2 - i - j)*((i + j - 1) mod 2)+(j^2 + (2*i - 1)*j + i^2 - i)/2 + (j - 1)*(1 - 2*((i + j) mod 2)) if i + j <= n + 1 and A[i,j] = n^2 - ((4*n^2 + (- 4*j - 4*i + 6)*n + j^2 + (2*i - 3)*j + i^2 - 3*i + 2)/2 + (i + j - 2*n)*((2*n - i - j + 1) mod 2)) + 1 - (n - j)*(1 - 2*((i + j) mod 2)) if i + j > n + 1 (proved). - Stefano Spezia, Aug 11 2018

			n=2, det=-2: {1 2 / 3 4 }
n=3, det=15: {1 2 6 / 3 5 7 / 4 8 9 }
n=4, det=-594: { 1 2 6 7 / 3 5 8 13 / 4 9 12 14 / 10 11 15 16 }
n=5, det=-5187: { 1 2 6 7 15 / 3 5 8 14 16 / 4 9 13 17 22 / 10 12 18 21 23 / 11 19 20 24 25 }

Vaclav Kotesovec, Table of n, a(n) for n = 1..1450
Index entries for linear recurrences with constant coefficients, signature (0,-9,0,-36,0,-84,0,-126,0,-126,0,-84,0,-36,0,-9,0,-1).

Cf. A023999, A069480, A079340.

A078475 := function(k)
local i, j, n;
for n in [1 .. k] do
   A:=NullMat(n,n);
   for i in [1 .. n] do
      for j in [1 .. n] do
         if i+j<=n+1 then
            A[i][j] := (2-i-j)*RemInt(i+j-1,2)+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*RemInt(i+j,2));;
         else
            A[i][j] := n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*RemInt(2*n-i-j+1,2))+1-(n-j)*(1-2*RemInt(i+j,2));
         fi;
      od;
   od;
   Print(n," ",Determinant(A),"\n");
od;
end;
A078475(27); # Stefano Spezia, Aug 12 2018

a[i_, j_, n_] := If[i+j<=n+1, (2-i-j)*Mod[i+j-1,2]+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*Mod[i+j,2]),n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*Mod[2*n-i-j+1,2])+1-(n-j)*(1-2*Mod[i+j,2])]; f[n_] := Det[ Table[a[i, j, n], {i, n}, {j, n}]]; Array[f, 27] (* Stefano Spezia, Aug 11 2018 *)

A(i,j,n) = if (i + j <= n + 1, (2 - i - j)*((i + j - 1) % 2)+(j^2 + (2*i - 1)*j + i^2 - i)/2 + (j - 1)*(1 - 2*((i + j) % 2)), n^2 - ((4*n^2 + (- 4*j - 4*i + 6)*n + j^2 + (2*i - 3)*j + i^2 - 3*i + 2)/2 + (i + j - 2*n)*((2*n - i - j + 1) % 2)) + 1 - (n - j)*(1 - 2*((i + j) % 2)));
a(n) = matdet(matrix(n, n, i, j, A(i, j, n))); \\ Michel Marcus, Aug 11 2018
(MATLAB, FreeMat and Octave)
for(n=1:27)
   A=zeros(n,n);
   for(i=1:n)
      for(j=1:n)
         if(i+j<=n+1)
            A(i,j)=(2-i-j)*mod(i+j-1,2)+(j^2+(2*i-1)*j+i^2-i)/2+(j-1)*(1-2*mod(i+j,2));
         else
            A(i,j)=n^2-((4*n^2+(-4*j-4*i+6)*n+j^2+(2*i-3)*j+i^2-3*i+2)/2+(i+j-2*n)*mod(2*n-i-j+1,2))+1-(n-j)*(1-2*mod(i+j,2));
         end
      end
   end
   fprintf('%d %0.f\n',n,det(A));
end # Stefano Spezia, Aug 12 2018

From Vaclav Kotesovec, Jan 08 2019: (Start)
Recurrence: (5*n^16 - 176*n^15 + 2888*n^14 - 29332*n^13 + 206454*n^12 - 1068276*n^11 + 4205934*n^10 - 12861022*n^9 + 30891328*n^8 - 58524140*n^7 + 87229074*n^6 - 101275380*n^5 + 89823673*n^4 - 58824210*n^3 + 26795412*n^2 - 7559784*n + 985608)*a(n) = 8*(n^14 - 20*n^13 + 169*n^12 - 754*n^11 + 1630*n^10 + 564*n^9 - 15184*n^8 + 52244*n^7 - 109015*n^6 + 167071*n^5 - 202816*n^4 + 191592*n^3 - 125145*n^2 + 45333*n - 5832)*a(n-1) - (5*n^16 - 96*n^15 + 848*n^14 - 4580*n^13 + 16966*n^12 - 45892*n^11 + 94310*n^10 - 151266*n^9 + 192520*n^8 - 195196*n^7 + 155666*n^6 - 94052*n^5 + 39329*n^4 - 6798*n^3 - 4572*n^2 + 5400*n - 1944)*a(n-2).
a(n) ~ ((-1)^n - 3) * (cos(Pi*n/2) + sin(Pi*n/2)) * n^8 / 72. (End)
a(n) = -9*a(n-2) - 36*a(n-4) - 84*a(n-6) - 126*a(n-8) - 126*a(n-10) - 84*a(n-12) - 36*a(n-14) - 9*a(n-16) - a(n-18) for n > 18. - Stefano Spezia, Apr 25 2021, simplified by Boštjan Gec, Sep 21 2023

Edited and extended by Robert G. Wilson v, May 08 2003

A126224 Determinant of the n X n matrix in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry.

Original entry on oeis.org

1, -5, -48, 660, 11760, -257040, -6652800, 198918720, 6745939200, -255826771200, -10727081164800, 492775291008000, 24610605962342400, -1327677426915840000, -76940526008586240000, 4766815315895592960000, 314406967644177408000000, -21995911456386651463680000

Offset: 1

Emeric Deutsch, Dec 31 2006

sign

			For n = 2, the 2 X 2 (spiral) matrix A is
      [1, 2]
      [4, 3]
Then a(2) = -5 because det(A) = 1*3 - 2*4 = -5.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Sergey Sadov, Problem 11270, American Mathematical Monthly, Vol. 114, No. 1, 2007, p. 78.

Cf. A023999 (absolute values). - Alois P. Heinz, Jan 21 2014

a:=n->(-1)^(n*(n-1)/2)*2^(2*n-3)*(3*n-1)*product(1/2+k,k=0..n-2): seq(a(n),n=1..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, (3*n+1)/4,
      4*(1-3*n)*(2*n-5)*(2*n-3) *a(n-2) /(3*n-7))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 21 2014

a[n_] := (-1)^(n*(n-1)/2)*2^(2n-3)*(3n-1)*Pochhammer[1/2, n-1]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, May 26 2015 *)

a(n) = (-1)^[n*(n-1)/2]*2^(2*n-3)*(3*n-1)*Product_{k=0..n-2} (1/2+k) for n>=2.

E.g.f.: (((-16*x^2-1)*sqrt(2*sqrt(16*x^2+1)+2)-8*sqrt(16*x^2+1)*x^2+16*x^2 + sqrt(16*x^2+1)+1)*sqrt(2*sqrt(16*x^2+1)-2)+(8*(sqrt(16*x^2+1)*x^2+2*x^2-(1/8) * sqrt(16*x^2+1)+1/8))*sqrt(2*sqrt(16*x^2+1)+2))/(512*x^3+32*x). - Robert Israel, Apr 20 2017

A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

Original entry on oeis.org

1, 5, 72, 1380, 31920, 861840, 26611200, 925404480, 35805369600, 1526139014400, 71066912716800, 3590219977344000, 195589552648089600, 11430978821982720000, 713448513897799680000, 47363888351558338560000

Offset: 1

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

If n == 0 or 1 (mod 4), the sign of the determinant will be independent of the orientation of the spiral. For n == 2 or 3 (mod 4), the sign will be reversed when the orientation is rotated by 1/4 or flipped on the horizontal or vertical axis. - Franklin T. Adams-Watters, Dec 31 2013

This distribution of the integers is sometimes known as Ulam's spiral, although that is sometimes reserved for when the primes are marked out in some way. - Franklin T. Adams-Watters, Dec 31 2013

			n=2, det=-5: {1 2 / 4 3 }.
n=3, det=72: {7 8 9 / 6 1 2 / 5 4 3 }.
n=4, det=-1380: { 7 8 9 10 / 6 1 2 11 /5 4 3 12 / 16 15 14 13 }.
n=5, det=31920: { 21 22 23 24 25 / 20 7 8 9 10 / 19 6 1 2 11 /18 5 4 3 12 / 17 16 15 14 13 }

Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.

Cf. A078475, A023999, A067276, A052182.

M[0, 0] = 1;
M[i_, j_] := If[i <= j,
  If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],
   M[i, j + 1] + 1],
  If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]
  ]
M[n_] := If[EvenQ[n],
  Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],
  Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]
a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *)

A079340(n):=if n=1 then 1 else (2*n^2-3*n+3)*(2*n-2)!/(2*(n-1)!)$
makelist(A079340(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n) = (2*n^2-3*n+3) (2n-2)!/(2 (n-1)!) = A096376(n-1)*A000407(n-2), n>1. - Conjectured by Dean Hickerson, Jan 30 2003. Proved in the article by Bhatnagar and Krattenthaler.

D-finite with recurrence (2*n^2-7*n+8)*a(n) -2*(2*n-3)*(2*n^2-3*n+3)*a(n-1)=0. - R. J. Mathar, May 03 2019

Extended by Robert G. Wilson v, Jan 25 2003

A226167 Array read by antidiagonals: a(i,j) is the number of ways of labeling a tableau of shape (i,1^j) with the integers 1, 2, ... i+j-2 (each label being used once) such that the first row is decreasing, and the first column has m-1 labels.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 27, 7, 1, 360, 168, 48, 9, 1, 2520, 1200, 360, 75, 11, 1, 20160, 9720, 3000, 660, 108, 13, 1, 181440, 88200, 27720, 6300, 1092, 147, 15, 1, 1814400, 887040, 282240, 65520, 11760, 1680, 192, 17, 1, 19958400, 9797760, 3144960, 740880, 136080, 20160, 2448, 243, 19, 1

Offset: 1

John M. Campbell, May 29 2013

For an arbitrary composition c, let F_c^p denote the linear transformation of NSym that is adjoint to multiplication by the fundamental quasi-symmetric function indexed by c. Then a(i,j) equals the coefficient of H_(1,1) in (F_(1)^p)^(i+j-2)(H_(i,1^j)) (see below SAGE program, and Corollary 2.7 in the below link).
Let M(n) = [a(i,j)]_{n x n}. Then det(M(n))=A000178(n)=the n-th superfactorial.
Let p_n(x) denote the polynomial such that a(x,n)=p_n(x). Then the coefficient of x in p_n(x) is |A009575(n)|. For example, p_4(x)=4x^3+18x^2+26x+12, and the coefficient of x in p_4(x) is |A009575(4)|=26.
First row is A001710. Second row is A138772. Fourth row is A136659.

			There are a(3,2) = 7 ways of labeling the tableau of shape (3,1,1) with 1, 2 and 3 (with each label being used once) such that the first row is decreasing and the first column has 1 label:
1    2    3    X    X    X    X
X    X    X    1    2    3    X
X32  X31  X21  X32  X31  X21  321
The matrix [a(i,j)]_(6 x 6) is given below:
[1  3  12   60   360   2520]
[1  5  27  168  1200   9720]
[1  7  48  360  3000  27720]
[1  9  75  660  6300  65520]
[1 11 108 1092 11760 136080]
[1 13 147 1680 20160 257040]

Alois P. Heinz, Rows n = 1..141, flattened
C. Berg, N. Bergeron, F. Saliola, L. Serrano, and M. Zabrocki, A Lift of the Schur and Hall-Littlewood Bases to Non-Commutative Symmetric Functions, 10-11.

Cf. A000178, A009575.

Main diagonal gives: A023999. - Alois P. Heinz, Jan 21 2014

a:= (i, j)-> (i+j-2)!/i!*(2*i+j-1)*j/2:
seq(seq(a(i, 1+d-i), i=1..d), d=1..12);  # Alois P. Heinz, Jan 21 2014

a[n_,k_]:=(n+k-2)!/n!*(2*n+k-1)*k/2 ;
Print[Array[a[#1,#2]&,{50,50}]//MatrixForm]
(* A program which gives a list of tableaux *)
a[i_, j_] :=  Module[{f, list1, el, emptylist, n},
  f[q_] := StringReplace[StringReplace[StringReplace[    StringReplace[ToString[q], ToString[i + j - 1] -> "X"], ", " -> ""], "{" -> ""], "}" -> ""]; list1 = Permutations[Join[Table[q, {q, 1, i + j - 2}], {i + j - 1, i + j - 1}]]; el[q_] := First[Take[list1, {q, q}]]; emptylist = {}; n = 1; While[n < 1 + Length[list1], If[Take[el[n], {j + 1, i + j}] == Sort[Take[el[n], {j + 1, i + j}], Greater] && Count[Take[el[n], {1, j + 1}], i + j - 1] == 2, emptylist = Append[emptylist, f[el[n]]], Null]; n++]; Print[emptylist]]

NSym = NonCommutativeSymmetricFunctions(QQ) ;
QSym = QuasiSymmetricFunctions(QQ) ;
F = QSym.Fundamental() ;
H = NSym.complete() ;
def a(n, m):
     expr = H([n]+[1 for q in range(m)]) ;
     w=1 ;
     while w

a(i,j) = (i+j-2)!/i!*(2*i+j-1)*j/2.

A376161 Number of support Tau-tilting modules for some algebras.

Original entry on oeis.org

3, 5, 12, 33, 98, 306, 990, 3289, 11154, 38454, 134368, 474810, 1693812, 6091780, 22064130, 80410185, 294647250, 1084922190, 4012165080, 14895504030, 55496654460, 207431394300, 777601790940, 2922867908298, 11013796950228, 41596652545756, 157434454904160, 597029454416724, 2268232385053096

Offset: 0

F. Chapoton, Sep 13 2024

nonn

See Prop. A.6 in Wang's reference for the table counting Tau-tilting modules for the linear quiver modulo the relation alpha*beta = 0.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Anna Rodriguez Rasmussen, Exact Borel subalgebras of quasi-hereditary monomial algebras, arXiv:2504.01706 [math.RT], 2025. See p. 38.
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019-2022.

Cf. A005807, A007054, A000108, A016789, A023999, A329533.

a := n -> -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2):
seq(a(n), n = 0..28)  # Peter Luschny, Sep 13 2024

A376161[n_] := CatalanNumber[n]*(9*n + 6)/(n + 2);
Array[A376161, 30, 0] (* Paolo Xausa, Sep 14 2024 *)

def a(n):
    return 3*(3*n+2)*binomial(2*n+4,n+2)/4/(2*n+1)/(2*n+3)

a(n) = 3*(3*n+2)*binomial(2*n+4,n+2)/(4*(2*n+1)*(2*n+3)).
a(n) = A329533(n)/(n + 1).
From Peter Luschny, Sep 13 2024: (Start)
a(n) = (3*n + 2) * [x^n] ((1 - 4*x)^(3/2) + 12*x - 2)/(4*x^2).
a(n) = A016789(n)*(3/2)*(2*n)! * [x^(2*n)] hypergeom([], [3], x^2).
a(n) = CatalanNumber(n)*(9*n + 6)/(n + 2).
a(n) = -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2).
a(n) = 2^n*(9*n + 6)*(2*n - 1)!! / (n + 2)!.
a(n) = A007054(n) * (3*n + 2) / 2.
a(n) = 6*A023999(n + 1)/(n + 2)!. (End)

cp's OEIS Frontend

A078475 Determinant of rank n matrix of 1..n^2 filled successively back and forth along antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

Extensions

A126224 Determinant of the n X n matrix in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A226167 Array read by antidiagonals: a(i,j) is the number of ways of labeling a tableau of shape (i,1^j) with the integers 1, 2, ... i+j-2 (each label being used once) such that the first row is decreasing, and the first column has m-1 labels.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A376161 Number of support Tau-tilting modules for some algebras.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula