A123967 - OEIS

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

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OFFSET

0,2

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

LINKS

Table of n, a(n) for n=0..54.

Eric Weisstein's World of Mathematics, Tridiagonal Matrix

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

EXAMPLE

Triangle starts:

-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:

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;

...

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 *)

PROG

(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

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).

Sequence in context: A294138 A207824 A179900 * A162259 A077195 A038243

Adjacent sequences: A123964 A123965 A123966 * A123968 A123969 A123970

KEYWORD

tabl,sign

AUTHOR

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

EXTENSIONS

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

STATUS

approved