A136390 - OEIS

%I #11 Jul 05 2019 21:01:32

%S 1,-4,2,6,-9,4,-4,16,-20,8,1,-14,41,-44,16,6,-44,102,-96,32,-1,26,

%T -129,248,-208,64,-8,96,-360,592,-448,128,1,-42,321,-968,1392,-960,

%U 256,10,-180,1002,-2528,3232,-2048,512,-1,62,-681,2972,-6448,7424,-4352,1024

%N Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,4}(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>=4 and k be of the same parity. Consider a set X consisting of (n+k)/2-4 blocks of the size 2 and an additional block of the size 4, then (-1)^((n-k)/2)a(n,k) is the number of n-4-subsets of X intersecting each block of the size 2.

%H Michael De Vlieger, <a href="/A136390/b136390.txt">Table of n, a(n) for n = 4..10194</a> (rows 4 <= n <= 200, flattened).

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic">Two enumerative functions</a>.

%H M. 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>=4 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-4, i)*binomial(n+k-4-2*i, n-4), i=0..(n+k)/2-4) and a(n,k)=0 if n and k are of different parity.

%e Rows are (1),(-4,2),(6,-9,4),(-4,16,-20,8),... since P_{4,4}=x^4, P_{5,4}=-4x^3+2x^5, P_{6,4}=6x^2-9x^4+4x^6,...

%p if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-4, i)*binomial(n+k-4-2*i, n-4), i=0..(n+k)/2-4); end if;

%t DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2) * Sum[(-1)^i * Binomial[(n + k)/2 - 4, i] Binomial[n + k - 4 - 2 i, n - 4], {i, 0, (n + k)/2 - 4}], {n, 4, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* _Michael De Vlieger_, Jul 05 2019 *)

%Y Cf. A008310, A053117.

%K sign,tabf

%O 4,2

%A _Milan Janjic_, Mar 30 2008, revised Apr 05 2008