|
|
A128503
|
|
Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.
|
|
4
|
|
|
1, 3, 6, -3, 10, -12, 15, -30, 6, 21, -60, 30, 28, -105, 90, -10, 36, -168, 210, -60, 45, -252, 420, -210, 15, 55, -360, 756, -560, 105, 66, -495, 1260, -1260, 420, -21, 78, -660, 1980, -2520, 1260, -168, 91, -858, 2970, -4620, 3150, -756, 28, 105, -1092, 4290, -7920, 6930, -2520, 252, 120, -1365
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
S2(n,x):=sum(S(n-k,x)*S1(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)) with the first convolution S1(n,x) given by array A128502.
Row polynomials P2(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).
|
|
LINKS
|
|
|
FORMULA
|
a(n,m)= binomial(n-m+2,2)*binomial(n-m,m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)= binomial(m+2,2)*binomial(n-m+2,m+2)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S2(n,x): 1/(1-x*z+z^2)^3.
G.f. for P2(n,x): 1/(1-z+x*z^2)^3
|
|
EXAMPLE
|
n=4: [15,-30,6] stands for the polynomial S2(4,x) = 15*x^4-30*x^2+6 = 2*(S(4,x)*S1(0,x)+S(3,x)*S1(1,x))+S(2,x)*S1(2,x).
n=4:[15,-30,6] stands also for the row polynomial P2(4,x) = 15-30*x+6*x^2.
[1];[3];[6,-3];[10,-12];[15,-30,6];[21,-60,30];[28,-105,90,-10];...
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,tabf,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|