A136388 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,2}(x) with 0 omitted (exponents in increasing order).

1, -2, 2, 1, -5, 4, 4, -12, 8, -1, 13, -28, 16, -6, 38, -64, 32, 1, -25, 104, -144, 64, 8, -88, 272, -320, 128, -1, 41, -280, 688, -704, 256, -10, 170, -832, 1696, -1536, 512, 1, -61, 620, -2352, 4096, -3328, 1024, 12, -292, 2072, -6400, 9728, -7168, 2048

Offset: 2

Author

Milan Janjic, Mar 30 2008, entry revised Apr 05 2008

Keywords

• sign
• tabf

Comments

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=2 and k be of the same parity. Consider a set X consisting of (n+k)/2-2 main blocks of the size 2 and an additional block of the size 2, then (-1)^((n-k)/2)a(n,k) is the number of n-2-subsets of X intersecting each main block.

Examples

			Rows are (1),(-2,2),(1,-5,4),(4,-12,8),(-1,13,-28,16),...
since P_{2,2}=x^2, P_{3,2}=-2x+2x^3, P_{4,2}=1-5x^2+4x^4,...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..10199 (rows 2 <= n <= 200, flattened).
• Milan Janjic, Two enumerative functions.
• M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Crossrefs

Cf. A008310, A053117.

Programs

Maple

if modp(n-k, 2)=0 then a[n,k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-2, i)*binomial(n+k-2-2*i, n-2), i=0..(n+k)/2-2); end if;

Mathematica

Rest@ Flatten@ Table[If[SameQ @@ Mod[{n, k}, 2], (-1)^((n - k)/2)*Sum[(-1)^i*Binomial[(n + k)/2 - 2, i]*Binomial[n + k - 2 - 2 i, n - 2], {i, 0, (n + k)/2 - 2}], 0], {n, 2, 13}, {k, Boole@ OddQ@ n, n, 2}] (* Michael De Vlieger, Jul 02 2019 *)

Formula

If n>=2 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-2, i)*binomial(n+k-2-2*i, n-2), i=0..(n+k)/2-2) and a(n,k)=0 if n and k are of different parity.