A052182 -id:A052182 - cp's OEIS frontend

A066917 Determinant of n X n matrix whose rows are cyclic permutations of 4..Composite(n).

4, -20, -216, 2025, 24457, -661745, -21930489, 485222400, 12094491106, -594800640896, -32850150466188, 1138086428486400, 42791463719713975, -3042278237035388153, -123027745203325414816, 5708406518410582200000, 275201338468042020170179

Offset: 1

Robert G. Wilson v, Jan 24 2002

			a(3) = -216 because this is the determinant of [ (4,6,8), (6,8,4), (8,4,6) ]

Cf. A052182, A002808.

Composite[ n_Integer ] := FixedPoint[ n + PrimePi[ # ] + 1 &, n + PrimePi[ n ] + 1 ]; f[ n_ ] := Module[ {a = Table[ Composite[ i ], {i, 1, n} ], m = {}, k = 0}, While[ k < n, m = Append[ m, RotateLeft[ a, k ] ]; k++ ]; Det[ m ] ]; Table[ f[ n ], {n, 1, 16} ]

A071210 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the degree k of the root.

Original entry on oeis.org

1, 3, 1, 18, 8, 1, 160, 80, 15, 1, 1875, 1000, 225, 24, 1, 27216, 15120, 3780, 504, 35, 1, 470596, 268912, 72030, 10976, 980, 48, 1, 9437184, 5505024, 1548288, 258048, 26880, 1728, 63, 1, 215233605, 127545840, 37200870, 6613488, 765450, 58320

Offset: 1

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.

Cf. A000312, A052182 (first column).

Maple
```
(n,k) -> binomial(n+1,k+1)*k*n^(n-k-1)
```

tabl(nn) = for (n=1, nn, for (k=1, n, print1(binomial(n+1, k+1)*k*n^(n-k-1), ", ");); print) \\ Michel Marcus, Jun 27 2013

T(n,k) = binomial(n+1, k+1)*k*n^(n-k-1).

A118702 a(n) = determinant of n X n circulant matrix whose first row is the first n Lucas numbers A000032, from L(0) to L(n-1).

Original entry on oeis.org

2, 3, 18, 0, 8347, -861952, 391524998, -359089453125, 893329160995712, -5499366235206395112, 87687141416511254851323, -3590079701896396800000000000, 381284797406693371431803926245802, -105147887074796935457211770823970013737

Offset: 1

Jonathan Vos Post, May 20 2006

			a(4) = 0 because of the singular matrix:
[2, 1, 3, 4]
[4, 2, 1, 3]
[3, 4, 2, 1]
[1, 3, 4, 2].

Eric Weisstein's World of Mathematics, Circulant Matrix.

A000032 Lucas numbers (beginning at 2): L(n) = L(n-1) + L(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. A000032, A048954, A052182, A066933, A086459, A086569.

circ[w_] := NestList[RotateRight, w, Length[w] - 1]; Table[ Det[ circ[ LucasL@ Range[0, n - 1]]], {n, 10}] (* Giovanni Resta, Jun 16 2016 *)

Corrected and extended by Giovanni Resta, Jun 16 2016

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A066917 Determinant of n X n matrix whose rows are cyclic permutations of 4..Composite(n).

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A071210 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the degree k of the root.

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A118702 a(n) = determinant of n X n circulant matrix whose first row is the first n Lucas numbers A000032, from L(0) to L(n-1).

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