A380661 -id:A380661 - cp's OEIS frontend

A384595 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -4, -18, -1059, -51115, -14122480, -5176201331, -8184762199782, -21582120875577408, -211126151053299550639, -3968236858233834575013603, -250193703665647266489840668160, -33362066597786815040358189976876663, -13879811335315653909400110618024123820786

Offset: 1

Clark Kimberling, Jul 10 2025

sign

			The rows of M(4) are (1,2,3,5), (5,1,2,3), (3,5,1,2), (2,3,5,1); determinant(M(4)) = -429; permanent(M(4)) = 1689, so neg(M(4)) = (-429 - 1689)/2 = -1059 and pos(M(4)) = (-429 + 1689)/2 = 630.

Cf. A000045, A118704 (determinant), A384594 (permanent), A384596.

z = 14;
v[n_] := Table[Fibonacci[k], {k, 2, n + 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384594 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A118704 *)
neg = (d - p)/2     (* A384595 *)
pos = (d + p)/2     (* A384596 *)

a(n) = (1/2)*(A118704(n) - A384594(n)).

A384596 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 36, 630, 75264, 10309104, 6689940744, 6609305472651, 25797682556382400, 181805125100075828614, 4497447436889592767655228, 225317753810449180044272832000, 36410024238337826166260377355303568, 12904889422278677354475665889049659231531

Offset: 1

Clark Kimberling, Jul 10 2025

nonn

			The rows of M(4) are (1,2,3,5), (5,1,2,3), (3,5,1,2), (2,3,5,1); determinant(M(4)) = -429; permanent(M(4)) = 1689, so neg(M(4)) = (-429 - 1689)/2 = -1059 and pos(M(4)) = (-429 + 1689)/2 = 630.

Cf. A000045, A118704 (determinant), A384594 (permanent), A384595.

z = 14;
v[n_] := Table[Fibonacci[k], {k, 2, n + 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384594 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A118704 *)
neg = (d - p)/2     (* A384595 *)
pos = (d + p)/2     (* A384596 *)

a(n) = (1/2)*(A118704(n) + A384594(n)).

A384592 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -1, -6, -150, -5370, -800640, -190224111, -176905217601, -291774510700416, -1732081174959242854, -20169113574187183899414, -780617739418078727245260288, -64339950490716309261029142530799, -16501635484912814610105139633118406681

Offset: 1

Clark Kimberling, Jul 02 2025

sign

			The rows of M(4) are (1,1,2,3), (3,1,1,2), (23,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 -265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384593.

z = 14;
v[n_] := Table[Fibonacci[k], {k, 1, n}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
Table[Permanent[u[n]], {n, 1, z}]   (* A384591  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
neg = (d - p)/2     (* A384592 *)
pos = (d + p)/2     (* A384593 *)

a(n) = (1/2)*((-1)^n A123745(n) - A384591(n)).

A384593 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 1, 10, 115, 7182, 630400, 236525784, 146492200737, 343945864714624, 1504008427530969535, 22752527891269251753118, 704885020930148344661403136, 70113811460119152088048405685808, 15355281759224121782879344275585373609

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

			The rows of M(4) are (1,1,2,3), (3,1,1,2), (2,3,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 - 265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384592.

z = 14;
v[n_] := Table[Fibonacci[k], {k, 1, n}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
Table[Permanent[u[n]], {n, 1, z}]   (* A384591  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
neg = (d - p)/2     (* A384592 *)
pos = (d + p)/2     (* A384593 *)

a(n) = (1/2)*((-1)^n A123745(n) + A384591(n)).

A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

1, 4, 225, 27728, 7240350, 3439197360, 2686774125000, 3213645578293248, 5578750547986764960, 13484491722080225280000, 43904082301794970311672000, 187409206411313292409598115840, 1025421491750171253824589270768000, 7056011383804251291488039375527526400

Offset: 1

Clark Kimberling, Mar 09 2025

nonn

			M(3) is the matrix with rows (1,2,3), (4,5,6), (7,8,9), determinant 0, permanent 450, negative part -225, and positive part 225.

Cf. A232773, A380661, A380662.

r[m_, n_] := Range[(m - 1) n + 1, m  n];
d = Table[Det[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
p = Table[Permanent[Table[r[m, n], {m, 1, n}]], {n, 1, 15}]
neg = (d - p)/2
pos = (d + p)/2

from sympy.functions.combinatorial.numbers import stirling, factorial
def A381723(n): return abs(sum(n**k*stirling(n,n-k,kind=1,signed=True)*stirling(n+1,k+1,kind=1,signed=True)*factorial(n-k)*factorial(k) for k in range(n+1)))>>1 if n>2 else 3*n-2 # Chai Wah Wu, Mar 25 2025

a(n) = A232773(n)/2 for n >= 3.

A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

Original entry on oeis.org

4, 16, 192, 3072, 61440, 1474560, 41287680, 1321205760, 47563407360, 1902536294400, 83711596953600, 4018156653772800, 208944145996185600, 11700872175786393600, 702052330547183616000, 44931349155019751424000, 3055331742541343096832000

Offset: 1

Clark Kimberling, Mar 05 2025

nonn

For a matrix M with determinant |M|, the numbers pos(M) and neg(M) are the positive and negative parts of |M|, as defined in A380661. The definition implies that (pos(M)+neg(M))/2 = |M| and (pos(M)-neg(M))/2 = permanent of M. Thus, M is singular if and only if pos(M) = - neg(M).
Guide to sequences pos(M(n)), where M(n) is the n X n matrix with every term = c, a constant:
  c = 1:  A001710
  c = 2:  A002866
  c = 3:  A032108
  c = 4:  this sequence
For each c >=1, let s(n) = pos(M(n)); then s(1) = c, s(2) = c^2, and s(n) = c*n*s(n-1) for n >= 3.

Cf. A001710, A002966, A032108, A380661.

Essentially the same as A051711.

c = 4; d = Table[Det[ConstantArray[c, {n, n}]], {n, 1, 18}]
p = Table[Permanent[ConstantArray[c, {n, n}]], {n, 1, 18}]
neg = (d - p)/2
pos = (d + p)/2

s(1) = 4, s(2) = 16, and s(n) = 4*n*s(n-1) for n >= 3.

A380649 Rectangular array ((-1)*D(i,j,1,2)) read by descending antidiagonals: D(i,j,s,n) denotes the determinant of the matrix described in Comments.

Original entry on oeis.org

1, 4, 3, 8, 7, 6, 13, 12, 11, 10, 19, 18, 17, 16, 15, 26, 25, 24, 23, 22, 21, 34, 33, 32, 31, 30, 29, 28, 43, 42, 41, 40, 39, 38, 37, 36, 53, 52, 51, 50, 49, 48, 47, 46, 45, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66

Offset: 1

Clark Kimberling, Jan 31 2025

Suppose that (m(i,j)) is a rectangular array of infinitely many rows and infinitely many columns. For integers s>=1 and n>=1, let M(i,j,s,n) be the nXn matrix (m(i+h*s,j+k*s)), where h=0..n-1, k=0..n-1.
Let D(i,j,s,n) and P(i,j,s,n) denote the determinant and permanent of M(i,j,s,n), respectively. For A380649, we take (m(i,j)) to be the natural number array (see A000027, A185787, and A144112), and ((-1)*D(i,j,1,2)) is as shown below in Example.
* D(i,j,1,1) = M(i,j,1,1) = m(i,j) has linearly recurrent row sequences, all with signature (3,-3,1).
* D(i,j,1,2) has linearly recurrent row sequences, all with signature (3,-3,1).
* (-1)*D(i,j,s,3) is the constant array in which every term is s^6, for all i,j,s.
* D(i,j,s,n) is the constant 0 array for all n>=4, for all i,j,s.
* P(i,j,s,n) depends only on n, and the rows all have the following linear recurrence signature:
b(2n+1,1), - b(2n+1,2), b(2n+1-3),..., -(2n+1,2n), 1, where b=binomial.
((-1)*D(i,j,1,2)) includes, exactly once, every positive integer not in A000096. The order array of ((-1)*D(i,j,1,2)) is the array in Example in A038722; see A333029 for the definition of order array.

			Corner of (-1)*D(i,j,1,2):
   1   4    8   13   19   26   34   43   53   64   76   89
   3   7   12   18   25   33   42   52   63   75   88  102
   6  11   17   24   32   41   51   62   74   87  101  116
  10  16   23   31   40   50   61   73   86  100  115  131
  15  22   30   39   49   60   72   85   99  114  130  147
  21  29   38   48   59   71   84   98  113  129  146  164
  28  37   47   58   70   83   97  112  128  145  163  182
  36  46   57   69   82   96  111  127  144  162  181  201
  45  56   68   81   95  110  126  143  161  180  200  221
  55  67   80   94  109  125  142  160  179  199  220  242
  66  79   93  108  124  141  159  178  198  219  241  264
  78  92  107  123  140  158  177  197  218  240  263  287
m(1,1) = 1, so M(1,1,1,2) is the matrix having (row 1) = (1,2) and (row 2) = (3,5), so (-1)*D(1,1,1,2) = -(1*5-2*3) = 1.

Cf. A000027, A000096, A038722, A185787, A333029, A380660, A380661.

s = 1; n = 2; z = 12;
r[n_, k_] := n + (n + k - 2)*(n + k - 1)/2;
Grid[Table[r[n, k], {n, 1, z}, {k, 1, z}]]  (* Array A000027 *)
FindLinearRecurrence[Table[r[1, k], {k, 1, 20}]]
t[i_, j_] := Table[r[i, j + k*s], {k, 0, n - 1}];
d[i_, j_] := -Det[Table[t[i + k*s, j], {k, 0, n - 1}]];  (* (-1)*D(i,j,s,n) *)
Grid[Table[d[i, j], {i, 1, z}, {j, 1, z}]]  (* array *)
FindLinearRecurrence[Table[d[12, k], {k, 1, 20}]]
Table[d[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* sequence *)

A380660 Rectangular array pos(i,j,1,2) read by descending antidiagonals: pos( ) and neg() denote the positive part and negative part of a determinant; see Comments.

Original entry on oeis.org

5, 16, 27, 48, 65, 84, 119, 144, 171, 200, 253, 288, 325, 364, 405, 480, 527, 576, 627, 680, 735, 836, 897, 960, 1025, 1092, 1161, 1232, 1363, 1440, 1519, 1600, 1683, 1768, 1855, 1944, 2109, 2204, 2301, 2400, 2501, 2604, 2709, 2816, 2925, 3128, 3243, 3360

Offset: 1

Clark Kimberling, Feb 04 2025

Suppose that (m(i,j)) is a rectangular array of infinitely many rows and infinitely many columns. For integers s>=1 and n>=1, let M(i,j,s,n) be the nXn matrix (m(i+h*s,j+k*s)), where h=0..n-1, k=0..n-1.
Let D(i,j,s,n) and P(i,j,s,n) denote the determinant and permanent of M(i,j,s,n), respectively. Define arrays pos(i,j,s,n) and neg(i,j,s,n) by pos(i,j,s,n) = (P(i,j,s,n)+D(i,j,s,n))/2 and neg(i,j,s,n) = (P(i,j,s,n)-D(i,j,s,n))/2, so that P(i,j,s,n) = pos(i,j,s,n)+neg(i,j,s,n) and D(i,j,s,n) = pos(i,j,s,n)-neg(i,j,s,n).
A definition of determinant of an nXn matrix (a(i,j)) is the sum of the products (-1)^p(u) a(1,j(1))*a(2,j(2))*...*a(n,j(n)) over the n! permutations u = (j(1),j(2),...,j(n)) of (1,2,...,n), where p(u) is the parity of u; i.e., p(u) = 0 or 1 according as u is an even or odd permutation; see Lang, pp. 452-3, especially Proposition 4.8.
We have:
  pos(i,j,s,n) is the sum of the n!/2 products for which p(u) = 0, and
  neg(i,j,s,n) is the sum of the n!/2 products for which p(u) = 1.
Here, the foundational array (m(i,j)) is the natural number array (see A000027, A185787, A144112). The row sequences of pos(i,j,s,n) and neg(i,j,s,n) are linearly recurrent with signature (5, -10, 10, -5, 1).

			Corner of pos(i,j,1,2):
     5     16     48    119    253    480    836   1363   2109
    27     65    144    288    527    897   1440   2204   3243
    84    171    325    576    960   1519   2301   3360   4756
   200    364    627   1025   1600   2400   3479   4897   6720
   405    680   1092   1683   2501   3600   5040   6887   9213
   735   1161   1768   2604   3723   5185   7056   9408  12319
  1232   1855   2709   3848   5332   7227   9605  12544  16128
  1944   2816   3975   5481   7400   9804  12771  16385  20736
  2925   4104   5632   7575  10005  13000  16644  21027  26245
  4235   5785   7752  10208  13231  16905  21320  26572  32763
  5940   7931  10413  13464  17168  21615  26901  33128  40404
  8112  10620  13699  17433  21912  27232  33495  40809  49288
M(1,1,1,2) is the matrix with (row 1) = (1,2), (row 2) =(3,5), so that
pos(1,1,1,2) = 1*5 = 5; neg(1,1,1,2) = 2*3 = 6; D(1,1,1,2) = -1; P(1,1,1,2) = 11.

S. Lang, Algebra, 2nd ed., Addison-Wesley, 1984, 452-453.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000027, A380649, A380661.

s = 1; n = 2; z = 12;
r[n_, k_] := n + (n + k - 2)*(n + k - 1)/2 (* Array A000027 *)
Grid[Table[r[n, k], {n, 1, z}, {k, 1, z}]]
t[i_, j_] := Table[r[i, j + k*s], {k, 0, n - 1}];
d[i_, j_] := Det[Table[t[i + k*s, j], {k, 0, n - 1}]];  (* D(i,j,s,n) *)
p[i_, j_] := Permanent[Table[t[i + k*s, j], {k, 0, n - 1}]];  (* P(i,j,s,n) *)
pos[i_, j_] := (p[i, j] + d[i, j])/2;
neg[i_, j_] := (p[i, j] - d[i, j])/2;
Grid[Table[pos[i, j], {i, 1, z}, {j, 1, z}]]  (* A380660 array *)
Grid[Table[neg[i, j], {i, 1, z}, {j, 1, z}]]  (* A380661 array *)
FindLinearRecurrence[Table[pos[1, k], {k, 1, 20}]] (* row recurrence, all rows *)
FindLinearRecurrence[Table[neg[7, k], {k, 1, 20}]] (* row recurrence, all rows *)
Table[pos[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380660 sequence *)
Table[neg[k, m - k], {m, 2, z}, {k, 1, m - 1}] // Flatten (* A380661 sequence *)

cp's OEIS Frontend

A384595 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A384596 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A384592 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A384593 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A381723 a(n) = pos(M(n)), where M(n) is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

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A380649 Rectangular array ((-1)*D(i,j,1,2)) read by descending antidiagonals: D(i,j,s,n) denotes the determinant of the matrix described in Comments.

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A380660 Rectangular array pos(i,j,1,2) read by descending antidiagonals: pos( ) and neg() denote the positive part and negative part of a determinant; see Comments.

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