A136674 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial p(n,x) with p(0,x) = 1, p(1,x) = 2 - x, p(2,x) = 1 - 4*x + x^2 and p(n,x) = (2-x)*p(n-1,x) - p(n-2,x) if n>2.

1, 2, -1, 1, -4, 1, 0, -8, 6, -1, -1, -12, 19, -8, 1, -2, -15, 44, -34, 10, -1, -3, -16, 84, -104, 53, -12, 1, -4, -14, 140, -258, 200, -76, 14, -1, -5, -8, 210, -552, 605, -340, 103, -16, 1, -6, 3, 288, -1056, 1562, -1209, 532, -134, 18, -1, -7, 20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1

Offset

0,2

Comments

Row sums: A117373(n-1).

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-1,k-1). - R. J. Mathar, Jan 12 2011

Example

Triangle begins as:
   1;
   2,  -1;
   1,  -4,   1;
   0,  -8,   6,    -1;
  -1, -12,  19,    -8,    1;
  -2, -15,  44,   -34,   10,    -1;
  -3, -16,  84,  -104,   53,   -12,    1;
  -4, -14, 140,  -258,  200,   -76,   14,   -1;
  -5,  -8, 210,  -552,  605,  -340,  103,  -16,   1;
  -6,   3, 288, -1056, 1562, -1209,  532, -134,  18,  -1;
  -7,  20, 363, -1848, 3575, -3640, 2170, -784, 169, -20, 1;

Maple

A136674aux := proc(n) option remember; if n = 0 then 1; elif n= 1 then 2-x ; elif n= 2 then 1-4*x+x^2 ; else (2-x)*procname(n-1)-procname(n-2) ; end if; end proc:
A136674 := proc(n, k) coeftayl(A136674aux(n), x=0, k) ; end proc: # R. J. Mathar, Jan 12 2011

Mathematica

(* tridiagonal matrix code*)
T[n_, m_, d_]:= If[n==m, 2, If[n==d && m==d-1, -3, If[(n==m-1 || n==m+1), -1, 0]]];
M[d_]:= Table[T[n, m, d], {n, d}, {m, d}];
Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]//Flatten
(* polynomial recursion: three initial terms necessary*)
p[x, 0]:= 1; p[x, 1]:= (2-x); p[x, 2]:= 1 -4*x +x^2;
p[x_, n_]:= p[x, n]= (2-x)*p[x, n-1] - p[x, n-2];
Table[ExpandAll[p[x, n]], {n, 0, Length[g] -1}]
(* Third program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, (-1)^n, If[k==0, 3-n, 2*T[n-1, k] -T[n-2, k] -T[n-1, k-1] ]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

Prog

(Sage)
@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return (-1)^n
    elif (k==0): return 3-n
    else: return 2*T(n-1, k) - T(n-2, k) - T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020

Crossrefs

Sequence in context: A248939 A106246 A340660 * A144383 A205553 A178411

Adjacent sequences: A136671 A136672 A136673 * A136675 A136676 A136677

Keyword

easy,tabl,sign

Author

Roger L. Bagula, Apr 05 2008

Status

approved