A207823 - OEIS

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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

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OFFSET

0,2

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

LINKS

Table of n, a(n) for n=0..54.

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.

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).

EXAMPLE

Triangle begins:

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:

0, 1

0, 4, 1

0, 15, 8, 1

0, 56, 46, 12, 1

0, 209, 232, 93, 16, 1

...

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 *)

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).

Sequence in context: A095307 A159764 A124029 * A056920 A123382 A197653

Adjacent sequences: A207820 A207821 A207822 * A207824 A207825 A207826

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Feb 20 2012

EXTENSIONS

Offset changed to 0 by Georg Fischer, Feb 18 2020

STATUS

approved