A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)).

1, 4, 10, -4, 20, -20, 35, -60, 10, 56, -140, 60, 84, -280, 210, -20, 120, -504, 560, -140, 165, -840, 1260, -560, 35, 220, -1320, 2520, -1680, 280, 286, -1980, 4620, -4200, 1260, -56, 364, -2860, 7920, -9240, 4200, -504, 455, -4004, 12870, -18480, 11550, -2520, 84, 560, -5460, 20020, -34320

Offset

0,2

Comments

S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m) with the second convolution S2(n,x) given by array A128503.

Row polynomials P3(n,x) := Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x).

Links

Table of n, a(n) for n=0..52.

Wolfdieter Lang, First 15 rows and more.

Formula

a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0.

a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0.

G.f. for S3(n,x): 1/(1-x*z+z^2)^4.

G.f. for P3(n,x): 1/(1-z+x*z^2)^4.

Example

  1;
  4;
  10,   -4;
  20,  -20;
  35,  -60,  10;
  56, -140,  60;
  84, -280, 210,  -20;
  120,-504, 560, -140;
  ...
n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2.

Crossrefs

Row sums (signed array) give A128506. Unsigned row sums are A001872.

Cf. A128503 (k=2 convolution).

Sequence in context: A151707 A059132 A059136 * A336988 A200454 A303052

Adjacent sequences: A128502 A128503 A128504 * A128506 A128507 A128508

Keyword

sign,tabf,easy

Author

Wolfdieter Lang, Apr 04 2007

Extensions

Name edited by Petros Hadjicostas, Sep 04 2019

Status

approved