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

Author

Philippe Deléham, Jan 28 2007

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
  ...
		

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.

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.

Programs

Magma

m:=12;
R:=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

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 *)

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

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)

Extensions

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