%I #20 Sep 26 2024 03:28:05
%S 1,-3,2,3,-7,4,-1,9,-16,8,-5,25,-36,16,1,-19,66,-80,32,7,-63,168,-176,
%T 64,-1,33,-192,416,-384,128,-9,129,-552,1008,-832,256,1,-51,450,-1520,
%U 2400,-1792,512,11,-231,1452,-4048,5632,-3840,1024,-1,73,-912,4424,-10496,13056,-8192,2048
%N Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,3}(x) with 0 omitted (exponents in increasing order).
%C 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).
%C Let n>=3 and k be of the same parity. Consider a set X consisting of (n+k)/2-3 blocks of the size 2 and an additional block of the size 3, then (-1)^((n-k)/2)a(n,k) is the number of n-3-subsets of X intersecting each block of the size 2.
%H Michael De Vlieger, <a href="/A136389/b136389.txt">Table of n, a(n) for n = 3..10197</a> (rows 3 <= n <= 200, flattened).
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic">Two enumerative functions</a>.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.2
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%F If n>=3 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-3, i)*binomial(n+k-3-2*i, n-3), i=0..(n+k)/2-3) and a(n,k)=0 if n and k are of different parity.
%e Rows are (1),(-3,2),(3,-7,4),(-1,9,-16,8),(-5,25,-36,16),...
%e since P_{3,3}=x^3, P_{4,3}=-3x^2+2x^4, P_{5,3}=3x-7x^3+4x^5,...
%p if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-3, i)*binomial(n+k-3-2*i, n-3), i=0..(n+k)/2-3); end if;
%t DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2)*Sum[(-1)^i*Binomial[(n + k)/2 - 3, i] Binomial[n + k - 3 - 2 i, n - 3], {i, 0, (n + k)/2 - 3}], {n, 3, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* _Michael De Vlieger_, Jul 05 2019 *)
%Y Cf. A008310, A053117.
%K sign,tabf
%O 3,2
%A _Milan Janjic_, Mar 30 2008, revised Apr 05 2008