|
%I #34 Jul 30 2018 03:31:46
%S 1,-5,1,24,-10,1,-115,73,-15,1,551,-470,147,-20,1,-2640,2828,-1190,
%T 246,-25,1,12649,-16310,8631,-2400,370,-30,1,-60605,91371,-58275,
%U 20385,-4225,519,-35,1,290376,-501150,374115,-157800,41140,-6790,693,-40,1,-1391275,2704755,-2313450,1142730,-359275,74571,-10220,892,-45,1
%N 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).
%C Riordan array (1/(1+5*x+x^2), x/(1+5*x+x^2)). - _Philippe Deléham_, Feb 03 2007
%C Chebyshev's S(n,x-5) polynomials (exponents of x in increasing order). - _Philippe Deléham_, Feb 22 2012
%C Row sums are A125905(n). - _Philippe Deléham_, Feb 22 2012
%C Diagonal sums are (-5)^n. - _Philippe Deléham_, Feb 22 2012
%C 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
%C Inverse of triangle in A125906. - _Philippe Deléham_, Feb 22 2012
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TridiagonalMatrix.html">Tridiagonal Matrix</a>
%F T(n,0) = (-1)^n*A004254(n+1).
%F G.f.: 1/(1+5*x+x^2 - y*x). - _Philippe Deléham_, Feb 22 2012
%F 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
%e Triangle starts:
%e 1;
%e -5, 1;
%e 24, -10, 1;
%e -115, 73, -15, 1;
%e 551, -470, 147, -20, 1;
%e -2640, 2828, -1190, 246, -25, 1;
%e 12649, -16310, 8631, -2400, 370, -30, 1;
%e ...
%e Triangle (0, -5, 1/5, -1/5, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
%e 1;
%e 0, 1;
%e 0, -5, 1;
%e 0, 24, -10, 1:
%e 0, -115, 73, -15, 1;
%e 0, 551, -470, 147, -20, 1;
%e 0, -2640, 2828, -1190, 246, -25, 1;
%e ...
%p 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
%t 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;
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 30 2018, after _Philippe Deléham_ *)
%o (Sage)
%o @CachedFunction
%o def A123967(n,k):
%o if n< 0: return 0
%o if n==0: return 1 if k == 0 else 0
%o return A123967(n-1,k-1)-A123967(n-2,k)-5*A123967(n-1,k)
%o for n in (0..9): [A123967(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y Cf. A123343, A004254.
%Y 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).
%K tabl,sign
%O 0,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 28 2006
%E Edited by _N. J. A. Sloane_, Dec 03 2006
|
