A086459 -id:A086459 - cp's OEIS frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368, -1562716604740038367719196682456673375

Offset: 1

Eric W. Weisstein

det(C(n)) = 0 for n divisible by 6.
The determinant of the circulant matrix is 0 when 6 divides n because the polynomial (x+1)^(6k) - 1 has roots that are roots of unity. See A086569 for a generalization. - T. D. Noe, Jul 21 2003
E. Lehmer claimed and J. S. Frame proved that 2^n - 1 divides a(n) and the quotient abs(a(n))/(2^n - 1) is a perfect square (Ribenboim 1999, p. 128). - Jonathan Sondow, Aug 17 2012
C(n) is the matrix whose first row is [c_1, ..., c_n] where c_i = binomial(n,i-1), and subsequent rows are obtained by cyclically shifting the previous row one place to the right: see examples and PARI code. - M. F. Hasler, Dec 17 2016

			a(2) = det [ 1 2 ; 2 1 ] = -3.
a(3) = det [ 1 3 3 ; 3 1 3 ; 3 3 1 ] = 28.
a(4) = det [ 1 4 6 4 ; 4 1 4 6 ; 6 4 1 4 ; 4 6 4 1 ] = -375.

P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.
P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023).

T. D. Noe, Table of n, a(n) for n=1..50
David W. Boyd, The asymptotic behaviour of the binomial circulant determinant, Journal of Mathematical Analysis and Applications, Volume 86, Issue 1, March 1982, Pages 30-38.
E. Brown and M. Chamberland, Generalizing Gauss's Gem, Amer. Math. Monthly, 119 (No. 7, 2012), 597-601. - _N. J. A. Sloane_, Sep 07 2012
D. Burde and W. A. Moens, The structure of Lie algebras with a derivation satisfying a polynomial identity, arXiv:2009.05434 [math.RA], 2020.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 10 (1959), 686-690.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 11 (1960), 730-733.
Joshua Cooper and Zhibin Du, Note on the spectra of Steiner distance hypermatrices, arXiv:2403.02287 [math.CO], 2024. See pp. 2, 4.
Greg Fee and Andrew Granville, The prime factors of Wendt's binomial circulant determinant, Math. Comp. 57 (1991), 839-848.
David Ford and Vijay Jha, On Wendt's Determinant and Sophie Germain's Theorem, Experimental Mathematics, 2 (1993) No. 2, 113-120.
J. S. Frame, Factors of the binomial circulant determinant, Fibonacci Quart., 18 (1980), pp. 9-23.
Charles Helou, On Wendt's Determinant, Math. Comp., 66 (1997) No. 219, 1341-1346.
Charles Helou and Guy Terjanian, Arithmetical properties of wendt's determinant, Journal of Number Theory, Volume 115, Issue 1, November 2005, Pages 45-57.
Emma Lehmer, On a resultant connected with Fermat's last theorem, Bull. Amer. Math. Soc. 41 (1935), 864-867.
Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114)
Anastasios Simalarides, Upper bounds for the prime divisors of Wendt's determinant, Math. Comp., 71 (2002), 415-427.
Eric Weisstein's World of Mathematics, Circulant matrix
E. Wendt, Arithmetische Studien über den letzten Fermatschen Satz, welcher aussagt, dass die Gleichung a^n=b^n+c^n für n>2 in ganzen Zahlen nicht auflösbar ist, Reimer (Berlin), 1894.

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A129205, A215615, A215616.

See A096964 for another definition.

a[ n_] := Resultant[ x^n - 1, (1+x)^n - 1, x];

{a(n) = if( n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n ))))}

a(n) = polresultant( x^n - 1, (1+x)^n - 1, x )

a(2*n) = A129205(n)^2 * (1-4^n).
a(n) = 0 if and only if 6 divides n. If d divides n, then a(d) divides a(n). - Michael Somos, Apr 03 2007
a(n) = (-1)^(n-1) * (2^n - 1) * A215615(n)^2. - Jonathan Sondow, Aug 17 2012
a(2*n) = -3 * A215616(n)^3. - Jonathan Sondow, Aug 18 2012

Additional comments from Michael Somos, May 27 2000 and Dec 16 2001

A086569 Product of the nonzero eigenvalues of the circulant matrix whose rows are formed by successively rotating a vector of binomial coefficients right. Generalization of A048954.

Original entry on oeis.org

1, -3, 28, -375, 3751, -49392, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, -67043511427995648000, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368

Offset: 1

T. D. Noe, Jul 21 2003

In sequence A048954, a determinant of a circulant matrix, a(n) = 0 when 6 divides n. The determinant of a matrix can be interpreted as the signed volume of a simplex whose vertices are given by the rows of the matrix. For n a multiple of 6, the points form a lower dimensional simplex that has zero volume in n-space. However, the volume in n-2 space is positive and is given by the product of the nonzero eigenvalues.

			a(6) = -49392 because -1, -28, -28 and 63 are the four nonzero eigenvalues of the matrix {{1,6,15,20,15,6}, {6,1,6,15,20,15}, {15,6,1,6,15,20}, {20,15,6,1,6,15}, {15,20,15,6,1,6}, {6,15,20,15,6,1}}.

For references, see A086459

Cf. A048954, A086459 (circulant of powers of 2).

Table[x=Binomial[n, Range[0, n-1]]; m=Table[RotateRight[x, i-1], {i, n}]; e=Eigenvalues[m]; prod=1; Do[If[e[[i]]!=0, prod=prod*e[[i]]], {i, n}]; FullSimplify[prod], {n, 15}]

A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

Original entry on oeis.org

4, -20, 361, -3567, 218053, -3455872, 736439027, -16245418225, 1519211613654, -37662452460912, 20199655476042865, -643524421698841536, 46513669467992431114, -3754367220494585505280, 277686193779526116536293, -123973821931125256333959105, 20103033234038999233385180658

Offset: 1

Jonathan Vos Post, May 20 2006

Semiprime analog of A066933 Circulant of prime numbers. a(n) alternates in sign. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A086459 Circulant of powers of 2.

			a(2) = -20 = determinant
|4,6|
|6,4|.
a(3) = 361 = 19^2 = determinant
|4,6,9|
|9,4,6|
|6,9,4|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A001358, A048954, A052182, A066933, A086459, A086569.

A118713 := proc(n)
    local C,r,c ;
    C := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
        C[r,c] := A001358(1+((c-r) mod n)) ;
    end do:
    end do:
    LinearAlgebra[Determinant](C) ;
end proc:
seq(A118713(n),n=1..13) ;

nmax = 13;
sp = Select[Range[3 nmax], PrimeOmega[#] == 2&];
a[n_] := Module[{M}, M[1] = sp[[1 ;; n]];
   M[k_] := M[k] = RotateRight[M[k - 1]];
   Det[Table[M[k], {k, 1, n}]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 16 2023 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A180602 a(n) = (2^(n+1) - 1)^n.

Original entry on oeis.org

1, 3, 49, 3375, 923521, 992436543, 4195872914689, 70110209207109375, 4649081944211090042881, 1227102111503512992112190463, 1291749870339606615892191271170049, 5429914198235566686555216227881787109375

Offset: 0

Paul D. Hanna, Sep 11 2010

More generally, we have the following identities:
(1) Sum_{n>=0} m^n* F(q^n*x)^b* log( F(q^n*x) )^n/n! = Sum_{n>=0} x^n* [y^n] F(y)^(m*q^n + b);
(2) Sum_{n>=0} m^n* q^(n^2)* exp(b*q^n*x)*x^n/n! = Sum_{n>=0} (m*q^n + b)^n*x^n/n! for all q, m, b.
This sequence results from (2) when q=2, m=2, b=-1.
For n >= 2, a(n) is the first number in a set of three powerful numbers in arithmetic progression with difference a(n)*(2^n - 1). - Arkadiusz Wesolowski, Aug 26 2013

			E.g.f: A(x) = 1 + 3*x + 7^2*x^2/2! + 15^3*x^3/3! + 31^4*x^4/4! +...
A(x) = exp(-x) + 2^2*exp(-2*x)*x + 2^6*exp(-4*x)*x^2/2! + 2^12*exp(-8*x)*x^3/3! +...

Cf. A086459 (signed, offset 1), variants: A055601, A079491, A136516, A165327.

Cf. A001694.

[(2^(n+1)-1)^n : n in [0..11]]; // Arkadiusz Wesolowski, Aug 26 2013

A180602:=n->(2^(n+1) - 1)^n: seq(A180602(n), n=0..10); # Wesley Ivan Hurt, Oct 09 2014

Table[(2^(n + 1) - 1)^n, {n, 0, 10}] (* Wesley Ivan Hurt, Oct 09 2014 *)

{a(n)=n!*polcoeff(sum(k=0, n, 2^(k^2+k)*exp(-2^k*x+x*O(x^n))*x^k/k!), n)}

def A180602(n): return ((1<Chai Wah Wu, Sep 13 2024

E.g.f.: Sum_{n>=0} 2^(n^2+n) * exp(-2^n*x) * x^n/n!.

Name changed by Arkadiusz Wesolowski, Aug 26 2013

A118704 a(n) = determinant of n X n circulant matrix whose first row is the first n distinct Fibonacci numbers A000045(2), A000045(3), ... A000045(n+1).

Original entry on oeis.org

1, -3, 18, -429, 24149, -3813376, 1513739413, -1575456727131, 4215561680804992, -29321025953223722025, 529210578655758192641625, -24875949855198086445567836160, 3047957640551011125902187378426905, -974921913036976554924444728974464589255

Offset: 1

Jonathan Vos Post, May 20 2006

a(n) alternates in sign.

			a(2) = -3 because of the determinant -3 =
| 1, 2 |
| 2, 1 |.
a(5) = 24149 = determinant
| 1, 2, 3, 5, 8 |
| 8, 1, 2, 3, 5 |
| 5, 8, 1, 2, 3 |
| 3, 5, 8, 1, 2 |
| 2, 3, 5, 8, 1 |.

Alois P. Heinz, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (<<0|1>, <1|1>>^(2+irem(n-i+j, n)))[1, 2])):
seq(a(n), n=1..15);  # Alois P. Heinz, Oct 23 2009

a(n) ~ (-1)^(n+1) * phi^(n*(n+1)) / 5^(n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 10 2025

Corrected and extended by Alois P. Heinz, Oct 23 2009

A118705 a(n) = determinant of n X n circulant matrix whose first row is the first n triangular numbers A000217(0), A000217(1), ... A000217(n-1).

Original entry on oeis.org

0, -1, 28, -1360, 105500, -12051585, 1908871832, -400855203840, 107838796034520, -36175347978515625, 14806446317943766420, -7263073394295238840320, 4206546078973080241293076, -2840250692354398785860048105, 2211476237421629752792968750000

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = - 1 because of the determinant -1 =
  | 0, 1 |
  | 1, 0 |.
a(4) = -1360 = determinant
  |0,1,3,6|
  |6,0,1,3|
  |3,6,0,1|
  |1,3,6,0|.

Robert Israel, Table of n, a(n) for n = 1..226
Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

f:= proc(n) uses LinearAlgebra;local i;
  Determinant(Matrix(n, shape=Circulant[[seq(i*(i+1)/2, i=0..n-1)]]))
end proc:
map(f, [$1..30]); # Robert Israel, Jan 25 2023

r[n_] := r[n] = Table[k(k+1)/2, {k, 0, n-1}];
M[n_] := Table[RotateRight[r[n], m-1], {m, 1, n}];
a[n_] := Det[M[n]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 11 2023 *)

a(n) = (-1)^(n-1)*n^(n-2)*(n+1)*(n-1)*((n+1)^n-(n-1)^n)/(6*2^n). [Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 03 2010]

More terms from Alois P. Heinz, Mar 16 2017

A118707 a(n) = determinant of n X n circulant matrix whose first row is the first n square numbers 0, 1, ..., (n-1)^2.

Original entry on oeis.org

0, -1, 65, -6720, 1080750, -252806400, 81433562119, -34630270976000, 18813448225370124, -12719917900800000000, 10478214213011739186685, -10333870908014534470926336, 12023263324381930168836397850, -16297888825404790818315505238016

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 65 = determinant
|0,1,4|
|4,0,1|
|1,4,0|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A000290 The squares: a(n) = n^2. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000290, A048954, A052182, A066933, A086459, A086569.

a(n) = (-1)^(n-1)*(n-1)*(2*n-1)*n^(n-2)*(n^n-(n-2)^n)/12 [From Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010]

More terms from Alois P. Heinz, Mar 16 2017

A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

Original entry on oeis.org

0, -1, 513, -532800, 1077540500, -3831689610000, 22051842087895137, -192710430555501494272, 2433436736207275231050384, -42684202683959414242500000000, 1007311823853329619224620155226025, -31149342348518897782279760206406615040

Offset: 1

Jonathan Vos Post, May 20 2006

			a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 513 = determinant
|0,1,8|
|8,0,1|
|1,8,0|.
a(6) = 22051842087895137 = determinant
|0,1,8,27,64,125,216|
|216,0,1,8,27,64,125|
|125,216,0,1,8,27,64|
|64,125,216,0,1,8,27|
|27,64,125,216,0,1,8|
|8,27,64,125,216,0,1|
|1,8,27,64,125,216,0|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

See also: A000578 The cubes: a(n) = n^3. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000578, A048954, A052182, A066933, A086459, A086569.

Table[Det[Table[RotateRight[Range[0,i]^3,n],{n,0,i}]],{i,0,10}] (* Harvey P. Dale, Oct 22 2012 *)

Contribution from Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010: (Start)
a(n) = (-1)^(n-1)*(n-1)^2*n^(n-2)*(n^(2n)-b(n)^n-c(n)^n+(n^2-3n+3)^n)/24
where
b(n)=(2*n^2-3*n-3+sqrt(15n^2-18n-9)i)/2 and
c(n)=(2*n^2-3*n-3-sqrt(15n^2-18n-9)i)/2 (End)

More terms from Harvey P. Dale, Oct 22 2012

A118712 a(n) = Determinant of n X n circulant matrix whose first row is A000001(1), A000001(2), ..., A000001(n) where A000001(n) = number of groups of order n.

Original entry on oeis.org

1, 0, 0, -5, 6, -16, 9, -134400, 647248, -1711908, 6076067, -85248000, 116477425, -1764364437, 909276004, -522319050599375232, 14313181351994538493, -165893335414907083200, 2939566160282258664451, -5007637771411479278976, 75399747694572065660672

Offset: 1

Jonathan Vos Post, May 20 2006

sign

			a(4) = -5 because of the determinant -5 =
|1,1,1,2|
|2,1,1,1|
|1,2,1,1|
|1,1,2,1|.
a(11) = 6076067 = determinant
|1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1|
|1, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2|
|2, 1, 1, 1, 1, 2, 1, 2, 1, 5, 2|
|2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 5|
|5, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1|
|1, 5, 2, 2, 1, 1, 1, 1, 2, 1, 2|
|2, 1, 5, 2, 2, 1, 1, 1, 1, 2, 1|
|1, 2, 1, 5, 2, 2, 1, 1, 1, 1, 2|
|2, 1, 2, 1, 5, 2, 2, 1, 1, 1, 1|
|1, 2, 1, 2, 1, 5, 2, 2, 1, 1, 1|
|1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 1|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A000001, A048954, A052182, A066933, A086459, A086569.

A118712 := n -> DeterminantMat(List([0..n-1], i->List([0..n-1], j->NrSmallGroups(((j-i) mod n)+1)))); # Eric M. Schmidt, Nov 17 2013

a(1) corrected by and more terms from Eric M. Schmidt, Nov 17 2013

A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n.

Original entry on oeis.org

1, -3, -8, 49, -24, -960, -48, -3375, 676, -8640, -120, -2247392, -168, -34560, -46080, 923521, -288, -28789488, -360, -54867456, -184320, -216000, -528, -89384770560, 15376, -423360, -512000, -438939648, -840, -558786571200, -960, -992436543, -1152000

Offset: 1

Rémy Sigrist, Feb 27 2019

From Robert Israel, Mar 06 2019: (Start)
a(n) is divisible by A000203(n).
If n is not a square, a(n) is divisible by A000203(n)*A071324(n).
(End)

			For n = 12:
- the divisors of 12 are: 1, 2, 3, 4, 6, 12,
- the corresponding circulant matrix is:
    [ 1 12  6  4  3  2]
    [ 2  1 12  6  4  3]
    [ 3  2  1 12  6  4]
    [ 4  3  2  1 12  6]
    [ 6  4  3  2  1 12]
    [12  6  4  3  2  1]
- its determinant is -2247392,
- hence, a(12) = -2247392.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Circulant matrix

Cf. A027750, A086459, A177894.

f:= proc(n) local F,d; uses numtheory, LinearAlgebra;
  F:= sort(convert(divisors(n),list));
  d:= nops(F);
  Determinant(Matrix(d,d,shape=Circulant[F]))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 06 2019

a[n_] := Module[{dd = Divisors[n], m, r}, m = Length[dd]; r = E^(2 Pi I/m); Product[Sum[dd[[j+1]] r^(j k), {j, 0, m-1}], {k, 0, m-1}] // FullSimplify];
Array[a, 100] (* Jean-François Alcover, Oct 17 2020 *)

a(n) = my (d=divisors(n)); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

Apparently, a(n) > 0 iff n is a square.
a(p) = p^2 - 1 for any prime number p.
a(p^2) = p^6 - 2*p^3 + 1 for any prime number p.
a(2^k) = A086459(k+1) for any k >= 0.
If p < q are primes, a(p*q) = -(p^4-1)*(q^2-1)^2. - Robert Israel, Mar 06 2019

cp's OEIS Frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

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A086569 Product of the nonzero eigenvalues of the circulant matrix whose rows are formed by successively rotating a vector of binomial coefficients right. Generalization of A048954.

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A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

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A180602 a(n) = (2^(n+1) - 1)^n.

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A118704 a(n) = determinant of n X n circulant matrix whose first row is the first n distinct Fibonacci numbers A000045(2), A000045(3), ... A000045(n+1).

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A118705 a(n) = determinant of n X n circulant matrix whose first row is the first n triangular numbers A000217(0), A000217(1), ... A000217(n-1).

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A118707 a(n) = determinant of n X n circulant matrix whose first row is the first n square numbers 0, 1, ..., (n-1)^2.

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