A125662 A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).

1, 3, 1, 8, 6, 1, 21, 25, 9, 1, 55, 90, 51, 12, 1, 144, 300, 234, 86, 15, 1, 377, 954, 951, 480, 130, 18, 1, 987, 2939, 3573, 2305, 855, 183, 21, 1, 2584, 8850, 12707, 10008, 4740, 1386, 245, 24, 1, 6765, 26195, 43398, 40426, 23373, 8715, 2100, 316, 27, 1

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)).

Equals A078812*A007318 as infinite lower triangular matrices.

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

Diagonal sums: A000244(powers of 3).

Row sums: A001353 (n+1).

Diagonals: A001906(n+1), A001871.

Cf. A000012, A008585, A062728.

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

Sequence in context: A030523 A207815 A123965 * A124025 A257488 A286416

Adjacent sequences: A125659 A125660 A125661 * A125663 A125664 A125665

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Jan 28 2007

Extensions

a(45) corrected and a(51) added by Philippe Deléham, Feb 19 2012

Status

approved