A128502 Convolution array for Chebyshev's S(n,x)=U(n,x/2) polynomials.

1, 2, 3, -2, 4, -6, 5, -12, 3, 6, -20, 12, 7, -30, 30, -4, 8, -42, 60, -20, 9, -56, 105, -60, 5, 10, -72, 168, -140, 30, 11, -90, 252, -280, 105, -6, 12, -110, 360, -504, 280, -42, 13, -132, 495, -840, 630, -168, 7, 14, -156, 660, -1320, 1260, -504, 56, 15, -182, 858, -1980, 2310, -1260, 252, -8, 16, -210, 1092

Offset

0,2

Comments

S1(n,x):=sum(S(n-k,x)*S(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)).

The unsigned column sequences, m>=0, divided by (m+1) give Pascal triangle column sequences for m+1.

G.f. for column m sequence: ((-1)^m)*(m+1)*(x^(2*m))/(1-x)^(m+2), m>=0.

Row polynomials P1(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).

Written as a triangle with increasing powers of x this is A294519. - Wolfdieter Lang, Nov 12 2017

Links

Table of n, a(n) for n=0..66.

W. Lang, First 15 rows and more.

Formula

a(n,m)=binomial(n-m,m)*(n+1-m)*(-1)^m, m=0..floor(n/2), n>=0.

a(n,m)=binomial(n+1-m,m+1)*(m+1)*(-1)^m, m=0..floor(n/2), n>=0.

G.f. for S1(n,x): 1/(1-x*z+z^2)^2.

G.f. for P1(n,x): 1/(1-z+x*z^2)^2.

Example

[1];[2];[3,-2],[4,-6];[5,-12,3];[6,-20,12];[7,-30,30,-4];[8,-42,60,-20];...
n=4: [5,-12,3] stands for the polynomial S1(4,x) = 5*x^4-12*x^2+3 = 2*(S(4,x)*1+S(3,x)*S(1,x))+S(2,x)*S(2,x).
n=4: [5,-12,3] stands also for the row polynomial P1(4,x) = 5-12*x+3*x^2.

Crossrefs

Row sums (signed array) give A099254. Unsigned row sums are A001629(n+2).

Cf. A115139 (with offset n>=0 is S(n, x) array, decreasing powers of x).

Cf. A294519 (as triangle).

Sequence in context: A328219 A328879 A130542 * A349382 A244306 A345199

Adjacent sequences: A128499 A128500 A128501 * A128503 A128504 A128505

Keyword

sign,tabf,easy

Author

Wolfdieter Lang Apr 04 2007

Status

approved