OFFSET
0,2
COMMENTS
Subtriangle of the triangle given by [0,3,-1/3,1/3,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A123965.
From Philippe Deléham, Feb 19 2012: (Start)
Riordan array (1/(1-3*x+x^2), x/(1-3*x+x^2)).
Triangle of coefficients of Chebyshev's S(n,x+3) polynomials (exponents of x in increasing order). (End)
For 1 <= k <= n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3} containing k-1 letters equal 3 and avoiding 01. - Milan Janjic, Dec 20 2016
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
FORMULA
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) - T(n-2,k); T(0,0)=1; T(n,k)=0 if k < 0 or k > n.
Sum_{k=0..n} T(n, k) = A001353(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000244(n+1).
G.f.: 1/(1-3*x+x^2-y*x). - Philippe Deléham, Feb 19 2012
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = abs( [x^k]( ChebyshevU(n, (3-x)/2) ) ).
Sum_{k=0..n} (-1)^k*T(n, k) = A000027(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000225(n). (End)
EXAMPLE
Triangle begins:
1;
3, 1;
8, 6, 1;
21, 25, 9, 1;
55, 90, 51, 12, 1;
...
Triangle [0,3,-1/3,1/3,0,0,0,...] DELTA [1,0,0,0,0,0,...] begins:
1;
0, 1;
0, 3, 1;
0, 8, 6, 1;
0, 21, 25, 9, 1;
0, 55, 90, 51, 12, 1;
...
MATHEMATICA
With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 3 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
Table[Abs[CoefficientList[ChebyshevU[n, (x-3)/2], x]], {n, 0, 12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)
PROG
(Magma)
m:=12;
R<x>:=PowerSeriesRing(Integers(), m+2);
A125662:= func< n, k | Abs( Coefficient(R!( Evaluate(ChebyshevU(n+1), (3-x)/2) ), k) ) >;
[A125662(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023
(SageMath)
def A125662(n, k): return abs( ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k] )
flatten([[A125662(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Jan 28 2007
EXTENSIONS
a(45) corrected and a(51) added by Philippe Deléham, Feb 19 2012
STATUS
approved