A294519 Convolution triangle for Chebyshev S polynomials (rising powers).

1, 0, 2, -2, 0, 3, 0, -6, 0, 4, 3, 0, -12, 0, 5, 0, 12, 0, -20, 0, 6, -4, 0, 30, 0, -30, 0, 7, 0, -20, 0, 60, 0, -42, 0, 8, 5, 0, -60, 0, 105, 0, -56, 0, 9, 0, 30, 0, -140, 0, 168, 0, -72, 0, 10, -6, 0, 105, 0, -280, 0, 252, 0, -90, 0, 11, 0, -42, 0, 280, 0, -504, 0, 360, 0, -110, 0, 12, 7, 0, -168, 0, 630, 0, -840, 0, 495, 0, -132, 0, 13

Offset: 0

Author

Wolfdieter Lang, Nov 07 2017

Keywords

• sign
• tabl
• easy

Comments

See the array A128502 without zeros and falling powers. This is the main entry.

The coefficient triangle for Chebyshev S polynomials is given in A049310.

The self-convolution (or first convolution) of the S polynomials is S1(n, x) := Sum_{k=0..n} S(k, x)*S(n-k, x), n >= 0, and S1(n, x) = Sum_{m=0..n} T(n, m)*x^m.

Examples

			The triangle T(n, m) begins:
n\m   0   1    2    3    4    5    6   7    8    9   10 11 12 ...
0:    1
1:    0   2
2:   -2   0    3
3:    0  -6    0    4
4:    3   0  -12    0    5
5:    0  12    0  -20    0    6
6:   -4   0   30    0  -30    0    7
7:    0 -20    0   60    0  -42    0   8
8:    5   0  -60    0  105    0  -56   0    9
9:    0  30    0 -140    0  168    0 -72    0   10
10:  -6   0  105    0 -280    0  252   0  -90    0   11
11:   0 -42    0  280    0 -504    0 360    0 -110    0 12
12:   7   0 -168    0  630    0 -840   0  495    0 -132  0 13
...
		

Crossrefs

Cf. A049310, A128502.

Formula

T(n, m) = [x^m] S1(n, x), with the first convolution S1 of the Chebyshev S polynomials. See a comment above.

T(n, m) = 0 if n-m is odd and T(n, m) = (-1)^((n-m)/2)*((n-m)/2 + 1)*binomial(n - (n-m)/2 +1, (n-m)/2 +1) = (-1)^((n-m)/2)*(n - (n-m)/2 + 1)* binomial(n - (n-m)/2, (n-m)/2) if n-m is even.

O.g.f. of {S1(n, x)}_{n >= 0} is G1(z,x) = (1/(1 - x*z + z^2))^2.