|
|
A159764
|
|
Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)).
|
|
13
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Row sums are (-1)^n*F(2n+2). Diagonal sums are (-1)^n*4^n. Inverse is A052179.
The positive matrix is (1/(1-4x+x^2), x/(1-4x+x^2)) with general term T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,2),0).
Triangle of coefficients of Chebyshev's S(n,x-4) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012
Subtriangle of 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, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012
|
|
LINKS
|
|
|
FORMULA
|
Number triangle T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,-2),0).
|
|
EXAMPLE
|
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;
Triangle (0, -4, 1/4, -1/4, 0, 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;
|
|
MATHEMATICA
|
CoefficientList[CoefficientList[Series[1/(1 + 4*x + x^2 - y*x), {x, 0, 10}, {y, 0, 10}], x], y]//Flatten (* G. C. Greubel, May 21 2018 *)
|
|
PROG
|
(Sage)
@CachedFunction
if n< 0: return 0
if n==0: return 1 if k == 0 else 0
|
|
CROSSREFS
|
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.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|