A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

1, 4, 1, 15, 8, 1, 56, 46, 12, 1, 209, 232, 93, 16, 1, 780, 1091, 592, 156, 20, 1, 2911, 4912, 3366, 1200, 235, 24, 1, 10864, 21468, 17784, 8010, 2120, 330, 28, 1, 40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1, 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1

Offset: 0

Author

Philippe Deléham, Feb 20 2012

Keywords

• easy
• nonn
• tabl

Comments

Riordan array (1/(1-4*x+x^2), x/(1-4*x+x^2)).

Subtriangle of the triangle given by (0, 4, -1/4, 1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Unsigned version of triangles in A124029 and in A159764.

For 1<=k<=n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3,4} containing k-1 letters equal 4 and avoiding 01. - Milan Janjic, Dec 20 2016

Examples

			Triangle begins:
  1
  4, 1
  15, 8, 1
  56, 46, 12, 1
  209, 232, 93, 16, 1
  780, 1091, 592, 156, 20, 1
  2911, 4912, 3366, 1200, 235, 24, 1
  10864, 21468, 17784, 8010, 2120, 330, 28, 1
  40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1
  151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
  ...
Triangle (0, 4, -1/4, 1/4, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 4, 1
  0, 15, 8, 1
  0, 56, 46, 12, 1
  0, 209, 232, 93, 16, 1
  ...
		

Links

• Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, J. Int. Seq. 21 (2018), #18.1.2.
• Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Crossrefs

Cf. Triangle of coefficients of 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).

Programs

Mathematica

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 4 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Formula

Recurrence: T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k).

Diagonal sums are 4^n = A000302(n).

Row sums are A004254(n+1).

G.f.: 1/(1-4*x+x^2-y*x)

T(n,n) = 1, T(n+1,n) = 4*n+4 = A008586(n+1), T(n+2,n) = (n+1)*(8n+15) = A139278(n+1).

T(n,0) = A001353(n+1).

Extensions

Offset changed to 0 by Georg Fischer, Feb 18 2020