A103247 Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere (n>=1). Row 0 consists of the single term 1.

1, -3, 1, 8, -6, 1, -20, 24, -9, 1, 48, -80, 48, -12, 1, -112, 240, -200, 80, -15, 1, 256, -672, 720, -400, 120, -18, 1, -576, 1792, -2352, 1680, -700, 168, -21, 1, 1280, -4608, 7168, -6272, 3360, -1120, 224, -24, 1, -2816, 11520, -20736, 21504, -14112, 6048, -1680, 288, -27, 1, 6144, -28160, 57600, -69120, 53760, -28224, 10080, -2400, 360, -30, 1

Offset

0,2

Comments

Row sums of the unsigned triangle yield A006234. The unsigned triangle is the mirror image of A103407.

Links

Table of n, a(n) for n=0..65.

Formula

Appears to be the matrix product (I-S)*P^(-2), where I is the identity, P is Pascal's triangle A007318 and S is A132440, the infinitesimal generator of P. Cf. A055137 (= (I-S)*P) and A103283 (= (I-S)*P^(-1)). - Peter Bala, Nov 28 2011

Example

The monic characteristic polynomial of the matrix [3 1 1 / 1 3 1 / 1 1 3] is x^3 - 9x^2 + 24x - 20; so T(3,0)=-20, T(3,1)=24, T(3,2)=-9, T(3,3)=1.
Triangle begins:
  1;
  -3,1;
  8,-6,1;
  -20,24,-9,1;
  48,-80,48,-12,1;
  ...

Maple

with(linalg): a:=proc(i, j) if i=j then 3 else 1 fi end: 1; for n from 1 to 10 do seq(coeff(expand(x*charpoly(matrix(n, n, a), x)), x^k), k=1..n+1) od; # yields the sequence in triangular form

Mathematica

M[n_] := Table[If[i == j, 3, 1], {i, 1, n}, {j, 1, n}];
P[n_] := P[n] = CharacteristicPolynomial[M[n], x];
row[n_] := row[n] = If[n == 0, {1}, CoefficientList[P[n]/Coefficient[P[n], x, n], x]];
T[n_, k_] := row[n][[k]];
Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 06 2024 *)

Crossrefs

Cf. A006234, A103407.

Sequence in context: A258018 A188939 A062196 * A030523 A207815 A123965

Adjacent sequences: A103244 A103245 A103246 * A103248 A103249 A103250

Keyword

sign,tabl

Author

Emeric Deutsch, Mar 19 2005

Status

approved