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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.

%I A103247 #16 Aug 06 2024 13:41:17
%S A103247 1,-3,1,8,-6,1,-20,24,-9,1,48,-80,48,-12,1,-112,240,-200,80,-15,1,256,
%T A103247 -672,720,-400,120,-18,1,-576,1792,-2352,1680,-700,168,-21,1,1280,
%U A103247 -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
%N 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.
%C A103247 Row sums of the unsigned triangle yield A006234. The unsigned triangle is the mirror image of A103407.
%F A103247 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
%e A103247 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.
%e A103247 Triangle begins:
%e A103247   1;
%e A103247   -3,1;
%e A103247   8,-6,1;
%e A103247   -20,24,-9,1;
%e A103247   48,-80,48,-12,1;
%e A103247   ...
%p A103247 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
%t A103247 M[n_] := Table[If[i == j, 3, 1], {i, 1, n}, {j, 1, n}];
%t A103247 P[n_] := P[n] = CharacteristicPolynomial[M[n], x];
%t A103247 row[n_] := row[n] = If[n == 0, {1}, CoefficientList[P[n]/Coefficient[P[n], x, n], x]];
%t A103247 T[n_, k_] := row[n][[k]];
%t A103247 Table[T[n, k], {n, 0, 10}, {k, 1, n+1}] // Flatten (* _Jean-François Alcover_, Aug 06 2024 *)
%Y A103247 Cf. A006234, A103407.
%K A103247 sign,tabl
%O A103247 0,2
%A A103247 _Emeric Deutsch_, Mar 19 2005