A123967 Triangle read by rows: T(0,0)=1; for n >= 1 T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the tridiagonal n X n matrix with main diagonal 5,5,5,... and sub- and superdiagonals 1,1,1,... (0 <= k <= n).

1, -5, 1, 24, -10, 1, -115, 73, -15, 1, 551, -470, 147, -20, 1, -2640, 2828, -1190, 246, -25, 1, 12649, -16310, 8631, -2400, 370, -30, 1, -60605, 91371, -58275, 20385, -4225, 519, -35, 1, 290376, -501150, 374115, -157800, 41140, -6790, 693, -40, 1, -1391275, 2704755, -2313450, 1142730, -359275, 74571, -10220, 892, -45, 1

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Oct 28 2006

Keywords

• tabl
• sign

Comments

Riordan array (1/(1+5*x+x^2), x/(1+5*x+x^2)). - Philippe Deléham, Feb 03 2007

Chebyshev's S(n,x-5) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012

Row sums are A125905(n). - Philippe Deléham, Feb 22 2012

Diagonal sums are (-5)^n. - Philippe Deléham, Feb 22 2012

Subtriangle of triangle given by (0, -5, 1/5, -1/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012

Inverse of triangle in A125906. - Philippe Deléham, Feb 22 2012

Examples

			Triangle starts:
      1;
     -5,      1;
     24,    -10,     1;
   -115,     73,   -15,     1;
    551,   -470,   147,   -20,   1;
  -2640,   2828, -1190,   246, -25,   1;
  12649, -16310,  8631, -2400, 370, -30, 1;
  ...
Triangle (0, -5, 1/5, -1/5, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,     1;
  0,    -5,    1;
  0,    24,  -10,     1:
  0,  -115,   73,   -15,   1;
  0,   551, -470,   147, -20,   1;
  0, -2640, 2828, -1190, 246, -25, 1;
  ...
		

Links

• Eric Weisstein's World of Mathematics, Tridiagonal Matrix

Crossrefs

Cf. A123343, A004254.

Cf. Chebyshev's S(n,x+k) polynomials : A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k=2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

Programs

Maple

with(linalg): m:=proc(i,j) if i=j then 5 elif abs(i-j)=1 then 1 else 0 fi end: T:=(n,k)->coeff(charpoly(matrix(n,n,m),x),x,k): 1; for n from 1 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] - 5 T[n-1, k] - T[n-2, k]; T[0, 0] = 1; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Philippe Deléham *)

Sage

@CachedFunction
def A123967(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A123967(n-1,k-1)-A123967(n-2,k)-5*A123967(n-1,k)
for n in (0..9): [A123967(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

T(n,0) = (-1)^n*A004254(n+1).

G.f.: 1/(1+5*x+x^2 - y*x). - Philippe Deléham, Feb 22 2012

T(n,k) = T(n-1,k-1) - 5*T(n-1,k) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 22 2014

Extensions

Edited by N. J. A. Sloane, Dec 03 2006