Peter J. C. Moses - cp's OEIS frontend

A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

0, 6, 7, 4, 1, 9, 2, 5, 9, 6, 9, 6, 7, 5, 6, 0, 7, 2, 5, 4, 7, 5, 3, 0, 6, 6, 6, 9, 2, 6, 7, 3, 0, 4, 6, 7, 1, 0, 1, 3, 0, 8, 6, 8, 9, 9, 9, 8, 9, 0, 1, 2, 8, 0, 8, 7, 2, 2, 2, 1, 2, 2, 4, 9, 1, 5, 0, 2, 5, 3, 5, 5, 4, 3, 6, 4, 6, 7, 3, 4, 1, 7, 4, 5, 9, 6, 2

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Minimum value of f"(x), where f(x) = (1 - 2^(1-x)) zeta(x), for 0 < x < 1:
0.0641392820642571684220887165127181687393656828446464013955957700...,
which occurs for x = 0.59737100658235275929541785444598... .

Cf. A113024, A367309, A367311.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

Original entry on oeis.org

0, 6, 4, 1, 3, 9, 2, 8, 2, 0, 6, 4, 2, 5, 7, 1, 6, 8, 4, 2, 2, 0, 8, 8, 7, 1, 6, 5, 1, 2, 7, 1, 8, 1, 6, 8, 7, 3, 9, 3, 6, 5, 6, 8, 2, 8, 4, 4, 6, 4, 6, 4, 0, 1, 3, 9, 5, 5, 9, 5, 7, 7, 0, 0, 2, 2, 5, 2, 5, 7, 6, 2, 7, 9, 8, 3, 6, 9, 3, 2, 1, 7, 2, 4, 9, 4, 7

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .

Cf. A113024, A367309, A367312.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

One initial 0 inserted by Artur Jasinski, Aug 04 2025

A294811 Let b(n) be the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are triangular numbers (A000217). Then a(n) = b(n)/A010551(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 6, 11, 16, 30, 48, 97, 157, 322, 524, 1077, 1777, 3684, 6157, 12876, 21684, 45520, 77212, 162533, 277608, 585993, 1006784, 2129433, 3677453, 7788711, 13514487, 28654668, 49933938, 105964856, 185377690, 393631445, 691101516, 1468137470

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Nov 09 2017

nonn

All terms are positive integers (for a proof, cf. comment in A293984). Note that a(1), a(2), a(3), a(4) remain the same if in the definition the triangular numbers are replaced by k-gonal numbers for k >= 5.

			Let n=3. For a permutation C={c_1,c_2,c_3}, set s = s(C) = c_1 - c_2 + c_3. We have the permutations:
1,2,3; s=2
1,3,2; s=0
2,1,3; s=4
2,3,1; s=0
3,1,2; s=4
3,2,1; s=2
Here there are 2 permutations for which {s} are triangular numbers (when s = 0). Further, since A010551(3) = 2, then a(3) = 1.
Let n=4. For a permutation C={c_1,c_2,c_3,c_4}, set s = s(C) = c_1 - c_2 + c_3 - c_4. We have the permutations:
1,2,3,4; s=-2
1,3,2,4; s=-4
2,1,3,4; s=0
2,3,1,4; s=-4
3,1,2,4; s=0
3,2,1,4; s=-2
1,2,4,3; s=0
1,3,4,2; s=0
2,1,4,3; s=2
2,3,4,1; s=2
3,1,4,2; s=4
3,2,4,1; s=4
1,4,2,3; s=-4
1,4,3,2; s=-2
2,4,1,3; s=-4
2,4,3,1; s=0
3,4,1,2; s=-2
3,4,2,1; s=0
4,1,2,3; s=2
4,1,3,2; s=4
4,2,1,3; s=0
4,2,3,1; s=4
4,3,1,2; s=0
4,3,2,1; s=2
Here there are 8 permutations for which {s} are triangular numbers (when s = 0). Further, since A010551(4) = 4, then a(4) = 8/4 = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (terms n=1..200 from Peter J. C. Moses)

Cf. A010551, A293857, A293984.

b:= proc(p, m, s) option remember; (n-> `if`(n=0, `if`(issqr(8*s+1), 1, 0),
      `if`(p>0, b(p-1, m, s+n), 0)+`if`(m>0, b(p, m-1, s-n), 0)))(p+m)
    end:
a:= n-> (t-> b(n-t, t, 0))(iquo(n, 2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 17 2020

polyQ[order_,n_]:=If[n==0,True,IntegerQ[(#-4+Sqrt[(#-4)^2+8 n (#-2)])/(2 (#-2))]&[order]];(*is a number polygonal?*)
Map[Total,Table[
possibleSums=Range[1/2-(-1)^n/2-Floor[n/2]^2,Floor[(n+1)/2]^2];
filteredSums=Select[possibleSums,polyQ[3,#]&&#>-1&];
positions=Map[Flatten[{#,Position[possibleSums,#,1]-1}]&,filteredSums];
Map[SeriesCoefficient[QBinomial[n,Floor[(n+1)/2],q],{q,0,#[[2]]/2}]&,positions],{n,25}]] (* Peter J. C. Moses, Jan 02 2018 *)

a(0)=1 prepended by Alois P. Heinz, Sep 17 2020

A294812 Let b(n) be the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are pentagonal numbers (A000326). Then a(n) = b(n)/A010551(n).

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 6, 10, 23, 38, 70, 110, 196, 346, 759, 1250, 2313, 3982, 8433, 14520, 29437, 50466, 102830, 179587, 376439, 654374, 1343540, 2352149, 4916286, 8654120, 18065200, 31783592, 66233160, 117371504, 246610521, 436972949, 913862320, 1626523783

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 09 2017

nonn

All terms are positive integers (for a proof, cf. comment in A293984).

Note that a(1), a(2), a(3), a(4) remain the same, if in the definition the pentagonal numbers are replaced by k-gonal numbers for k >= 3 other than k=4.

Peter J. C. Moses, Table of n, a(n) for n = 1..200

Cf. A010551, A293857, A293984, A294811.

polyQ[order_,n_]:=If[n==0,True,IntegerQ[(#-4+Sqrt[(#-4)^2+8 n (#-2)])/(2 (#-2))]&[order]];(*is a number polygonal?*)
Map[Total,Table[
possibleSums=Range[1/2-(-1)^n/2-Floor[n/2]^2,Floor[(n+1)/2]^2];
filteredSums=Select[possibleSums,polyQ[5,#]&&#>-1&];
positions=Map[Flatten[{#,Position[possibleSums,#,1]-1}]&,filteredSums];
Map[SeriesCoefficient[QBinomial[n,Floor[(n+1)/2],q],{q,0,#[[2]]/2}]&,positions],{n,25}]] (* Peter J. C. Moses, Jan 02 2018 *)

A293783 Triangle of numbers of squares {i^2}, i = 0,1..ceiling(n/2), in permutations of {1..n} in A293857.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 8, 0, 4, 0, 24, 0, 12, 0, 108, 0, 36, 576, 0, 720, 0, 144, 4608, 0, 4032, 0, 576, 0, 31680, 0, 31680, 0, 2880, 0, 288000, 0, 201600, 0, 14400, 2505600, 0, 2764800, 0, 1987200, 0, 86400, 30067200, 0, 28512000, 0, 14515200, 0, 518400

Offset: 1

Peter J. C. Moses and Vladimir Shevelev, Oct 18 2017

From Shevelev's comment in A008794 it follows that the last entries of rows, corresponding to the maximal possible i^2, form sequence A010551, n >= 1. Also note that the last entry of each row is the gcd of all its entries (for a proof see comment in A293857). - Vladimir Shevelev, Oct 26 2017

			Triangle begins
         0,      1;
         0,      1;
         2,      0,        2;
         8,      0,        4;
         0,     24,        0,     12;
         0,    108,        0,     36;
       576,      0,      720,      0,      144;
      4608,      0,     4032,      0,      576;
         0,  31680,        0,  31680,        0,  2880;
         0, 288000,        0, 201600,        0, 14400;
   2505600,      0,  2764800,      0,  1987200,     0,  86400;
  30067200,      0, 28512000,      0, 14515200,     0, 518400;
The compressed triangle resulting from the division of each entry by the last entry of its row begins as follows. If i is the index of the row, starting with i = 1 then this last entry is floor(i/2)! * (i - floor(i/2))!.
    0,   1;
    0,   1;
    1,   0,   1;
    2,   0,   1;
    0,   2,   0,   1;
    0,   3,   0,   1;
    4,   0,   5,   0,   1;
    8,   0,   7,   0,   1;
    0,  11,   0,  11,   0,   1;
    0,  20,   0,  14,   0,   1;
   29,   0,  32,   0,  23,   0,   1;
   58,   0,  55,   0,  28,   0,   1;
    0,  88,   0,  94,   0,  46,   0,   1;
    0, 169,   0, 146,   0,  53,   0,   1;
  263,   0, 282,   0, 283,   0,  86,   0,   1;
  526,   0, 515,   0, 383,   0,  97,   0,   1;
For the sense of the entries of this triangle see the [Shevelev] link (with a continuation there). Let B(n,i) be the set of permutations C of 1..n for which c_1 - c_2 + ... + (-1)^(n-1)*c_n = i^2, i >= 0. Then |B(n,i)| is the entry in the n-th row and i-th column of the first triangle. Let us call two permutations C_1 and C_2 equivalent if one of them is obtained from another by a permutation of its elements with odd indices and/or separately with even indices. Let b(n,i) be the entry in the n-th row and i-th column of the second triangle. Then b(n,i) is the maximal possible number of pairwise non-equivalent permutations which could be chosen in B(n,i). On the other hand, it is the smallest number of non-equivalent permutations in B(n,i) such that every other permutation in B(n,i) is equivalent to one of them. So in some sense b(n,i) is the dimension of B(n,i). In particular, b(n,i) = 0 corresponds to empty B(n,i). - _Vladimir Shevelev_, Nov 13 2017

Peter J. C. Moses, Table of the first 100 rows
Vladimir Shevelev, Basis in subsets of permutations described in A293783, SeqFan post Oct 27 2017.

Cf. A008794, A010551, A293857, A293984.

a293783=Flatten[Table[PadLeft[Riffle[#,Table[0,{Floor[(n-1)/4]}]/.{}->0],1+Floor[(1+n)/2]](Floor[n/2]!*(n-Floor[n/2])!)&[Reverse[Map[SeriesCoefficient[QBinomial[n,Floor[(n+1)/2],q],{q,0,#}]&,Map[2#(Floor[(n+1)/2] - #)&,Range[0,Floor[(n+1)/4]]]]]],{n,20}]] (* Peter J. C. Moses, Nov 01 2017 *)

Row sums of triangle give A293857.

If C = {c_1..c_n} is a permutation of {1..n}, then c_1 - c_2 + ... has the same parity as 1 + 2 + ... + n = n*(n+1)/2. So adjacent rows in the triangle for odd and even n have the same positions of 0's. These positions follow through one, beginning from the first position for n == 1,2 (mod 4) and from the second position for n == 3,0 (mod 4). - David A. Corneth and Vladimir Shevelev, Oct 19 2017

A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 10*k+j-1), for j=1..10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2276485387658524, -523547340003805770400, -39617190432735671861429500, -2896792542975174202888623380000, -95819032881785191861991031568287500, -1018409199709889673458815786392849200000

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

sign

a(n) = 0 for n == 9 (mod 10).

A generalization. For an even N >= 2, consider the determinant of circulant matrix of order N with entries in the first row (-1)^(j-1)K_j(n), j=1..N, where K_j(n) = Sum_{k>=0} (-1)^k*binomial(n, N*k+j-1). Then it is 0 for n == N-1 (mod N). This statement follows from an easily proved identity K_j(N*t + N - 1) = (-1)^t*K_(N - j + 1)(N*t + N - 1) and a known calculation formula for the determinant of circulant matrix [Wikipedia]. Besides, it is 0 for n=1..N-2. We also conjecture that every such sequence contains infinitely many blocks of N-1 negative and N-1 positive terms separated by 0's.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A290286, A290535, A290539.

f:= n -> LinearAlgebra:-Determinant(Matrix(10,10,shape=
  Circulant[seq((-1)^j*add((-1)^k*binomial(n,10*k+j),
     k=0..(n-j)/10), j=0..9)])):
map(f, [$0..20]); # Robert Israel, Aug 08 2017

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 10k+j-1], {k, 0, n/10}], {j, 1, 10}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 9}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Aug 10 2018 *)

A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 6*k + j - 1), j=1..6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -104284, -783050688, -329029322076, -43271152876224, -2175830808446736, 0, 5427970251634650916, 307609249050423946080, 8866068073884849492756, 137518739026000524646272, 896278292839676023110288, 0, -2518571790589921864549097500

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

sign

a(n) = 0 for n == 5 (mod 6).

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A290285, A290286.

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k Binomial[n, 6k+j-1], {k, 0, n/6}], {j, 1, 6}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 5}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 10 2018 *)

A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^kbinomial(n,8k+j-1), for j=1..8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, -8489565952, -31872959692800, -932158289501356032, -4169183582652459909120, -5144394740685202662359040, -2505627397073121215653085184, -500556279165026162974748835840, 0, 20396260728315877590754520243175424

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Aug 05 2017

sign

a(n) = 0 for n == 7 (mod 8).

Robert Israel, Table of n, a(n) for n = 0..428
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A290285, A290286, A290535, A290540.

seq(LinearAlgebra:-Determinant(Matrix(8,8,shape=Circulant[seq(
(-1)^(j-1)*add((-1)^k*binomial(n,8*k+j-1),k=0..n/8),j=1..8)])), n=0..20); # Robert Israel, Aug 11 2017

ro[n_] := Table[(-1)^(j-1) Sum[(-1)^k*Binomial[n, 8k+j-1], {k, 0, n/8}], {j, 1, 8}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 7}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 10 2018 *)

A290286 Determinant of circulant matrix of order 4 with entries in the first row (-1)^jSum_{k>=0}(-1)^kbinomial(n, 4*k+j), j=0,1,2,3.

Original entry on oeis.org

1, 0, 0, 0, -1008, -37120, -473600, 0, 63996160, 702013440, 2893578240, 0, -393379835904, -12971004067840, -160377313820672, 0, 21792325059543040, 239501351489372160, 987061897553510400, 0, -134124249770961666048, -4422152303189489090560

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jul 26 2017

sign

In the Shevelev link the author proved that, for odd N>=3 and every n>=1, the determinant of circulant matrix of order N with entries in the first row (-1)^j*Sum{k>=0}(-1)^k*binomial(n, N*k+j), j=0..N-1, is 0.

This sequence shows what happens for the first even N>3.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A099586 (prefixed by a(0)=1), A099587, A099588, A099589, A290285.

seq(LinearAlgebra:-Determinant(Matrix(4,shape=Circulant[seq((-1)^j*
add((-1)^k*binomial(n, 4*k+j),k=0..n/4),j=0..3)])),n=0..50); # Robert Israel, Jul 26 2017

ro[n_] := Table[Sum[(-1)^(j+k) Binomial[n, 4k+j], {k, 0, n/4}], {j, 0, 3}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 3}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

from sympy.matrices import Matrix
from sympy import binomial
def mj(j, n): return (-1)**j*sum((-1)**k*binomial(n, 4*k + j) for k in range(n//4 + 1))
def a(n):
    m=Matrix(4, 4, lambda i,j: mj((i-j)%4,n))
    return m.det()
print([a(n) for n in range(22)]) # Indranil Ghosh, Jul 31 2017

a(n) = 0 for n == 3 (mod 4).

G.f. (empirical): (1/8)*(68*x^2+1)/(16*x^4+136*x^2+1)+(1/4)*(68*x^2-8*x+1)/(16*x^4+64*x^3+128*x^2-16*x+1)+(1/2)*(12*x^2+1)/(16*x^4+24*x^2+1)+3/(8*(4*x^2+1))-(1/4)*(12*x^2-4*x+1)/(16*x^4-32*x^3+32*x^2-8*x+1)-(1/4)*(4*x^2+1)/(16*x^4+1)+(1/4)*(12*x^2+4*x+1)/(16*x^4+32*x^3+32*x^2+8*x+1). - Robert Israel, Jul 26 2017

A282594 Primes p > 5 such that odd part of (p^2-q^2)/3 is composite for every prime q, 3 < q < p.

Original entry on oeis.org

307, 353, 409, 461, 499, 509, 593, 647, 673, 743, 811, 863, 929, 1051, 1123, 1163, 1201, 1217, 1279, 1453, 1553, 1657, 1697, 1783, 1823, 1889, 1907, 1931, 1973, 2029, 2089, 2131, 2141, 2203, 2243, 2267, 2297, 2311, 2411, 2417, 2531, 2579, 2593, 2609, 2617

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 19 2017

nonn

If prime(n) is in the sequence, then necessarily A282445(n) = 0. On the other hand, if A282445(n) = 0, then prime(n) is in the sequence if and only if all numbers {odd part of (prime(n)^2-q^2)/3, q is prime, 3 < q < prime(n)} are more than 1.

			The smallest n for which A282445(n)=0 is 44. Prime(44)=193. For q=5,7,..., 181, odd part of (p^2-q^2)/3 is 4653,775,...,187 respectively which are all composite numbers. But for q=191, we have 1. Therefore, 193 is not in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000265, A282445.

is(n)=if(!isprime(n), return(0)); my(p2=n^2,t); forprime(q=5,n-2, t=(p2-q^2)/3; t>>=valuation(t,2); if(isprime(t) || t==1, return(0))); n > 5 \\ Charles R Greathouse IV, Feb 20 2017

cp's OEIS Frontend

User: Peter J. C. Moses

A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

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A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

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A294811 Let b(n) be the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are triangular numbers (A000217). Then a(n) = b(n)/A010551(n).

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A294812 Let b(n) be the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are pentagonal numbers (A000326). Then a(n) = b(n)/A010551(n).

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A293783 Triangle of numbers of squares {i^2}, i = 0,1..ceiling(n/2), in permutations of {1..n} in A293857.

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A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)*Sum_{k>=0} (-1)^k*binomial(n, 10*k+j-1), for j=1..10.

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Mathematica

A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)*Sum_{k>=0} (-1)^k*binomial(n, 6*k + j - 1), j=1..6.

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Mathematica

A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^k*binomial(n,8*k+j-1), for j=1..8.

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A290540 Determinant of circulant matrix of order 10 with entries in the first row that are (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 10*k+j-1), for j=1..10.

A290535 Determinant of circulant matrix of order 6 with entries in the first row (-1)^(j-1)Sum_{k>=0} (-1)^kbinomial(n, 6*k + j - 1), j=1..6.

A290539 Determinant of circulant matrix of order eight with entries in the first row that are (-1)^(j-1) * Sum_{k>=0} (-1)^kbinomial(n,8k+j-1), for j=1..8.

A290286 Determinant of circulant matrix of order 4 with entries in the first row (-1)^jSum_{k>=0}(-1)^kbinomial(n, 4*k+j), j=0,1,2,3.