A207824 Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).

1, 5, 1, 24, 10, 1, 115, 73, 15, 1, 551, 470, 147, 20, 1, 2640, 2828, 1190, 246, 25, 1, 12649, 16310, 8631, 2400, 370, 30, 1, 60605, 91371, 58275, 20385, 4225, 519, 35, 1, 290376, 501150, 374115, 157800, 41140, 6790, 693, 40, 1

Offset

0,2

Comments

Riordan array (1/(1-5*x+x^2), x/(1-5*x+x^2)).

Subtriangle of triangle given by (0, 5, -1/5, 1/5, 0, 0, 0, 0, 0, 0, 0, 0...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Unsigned version of A123967 and A179900.

For 1<=k<=n, T(n,k) equals the number of (n-1)-length words over {0,1,2,3,4,5} containing k-1 letters equal 5 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

Recurrence : T(n,k) = 5*T(n-1,k) + T(n-1,k-1) - T(n-2,k).

G.f.: 1/(1-5*x+x^2-y*x).

Diagonal sums are 5^n = A000351(n).

Row sums are A001109(n+1).

T(n,0) = A004254(n+1), T(n+1,n) = 5n+5 = A008587(n+1).

Example

Triangle begins :
  1
  5, 1
  24, 10, 1
  115, 73, 15, 1
  551, 470, 147, 20, 1
  2640, 2828, 1190, 246, 25, 1
  12649, 16310, 8631, 2400, 370, 30, 1
  ...
Triangle (0, 5, -1/5, 1/5, 0, 0, 0,...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 5, 1
  0, 24, 10, 1
  0, 115, 73, 15, 1
  0, 551, 470, 147, 20, 1
  0, 2640, 2828, 1190, 246, 25, 1
  ...

Mathematica

With[{n = 8}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 5 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Prog

(PARI) row(n) = Vecrev(polchebyshev(n, 2, (x+5)/2)); \\ Michel Marcus, Apr 26 2018

Crossrefs

Cf. Triangles 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).

Sequence in context: A146675 A201884 A294138 * A179900 A123967 A162259

Adjacent sequences: A207821 A207822 A207823 * A207825 A207826 A207827

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 20 2012

Status

approved