A128503 Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.

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

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

Table of n, a(n) for n=0..57.

W. Lang, First 13 rows and more.

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

Row sums (signed array) give A128504. Unsigned row sums are A001628.

Cf. A128502 (k=1 convolution). A128505 (k=3 convolution).

Sequence in context: A134440 A129529 A298263 * A210193 A368952 A120906

Adjacent sequences: A128500 A128501 A128502 * A128504 A128505 A128506

Keyword

sign,tabf,easy

Author

Wolfdieter Lang Apr 04 2007

Status

approved