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Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,5}(x) with 0 omitted (exponents in increasing order).
1

%I #11 Jul 05 2019 21:01:39

%S 1,-5,2,10,-11,4,-10,25,-24,8,5,-30,61,-52,16,-1,20,-85,146,-112,32,

%T -7,70,-231,344,-240,64,1,-34,225,-608,800,-512,128,9,-138,681,-1560,

%U 1840,-1088,256,-1,52,-501,1970,-3920,4192,-2304,512

%N Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,5}(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>=5 and k be of the same parity. Consider a set X consisting of (n+k)/2-5 blocks of the size 2 and an additional block of the size 5, then (-1)^((n-k)/2)a(n,k) is the number of n-5-subsets of X intersecting each block of the size 2.

%H Michael De Vlieger, <a href="/A136397/b136397.txt">Table of n, a(n) for n = 5..10190</a> (rows 5 <= 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>=5 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-5, i)*binomial(n+k-5-2*i, n-5), i=0..(n+k)/2-5) and a(n,k)=0 if n and k are of different parity.

%e Rows are (1),(-5,2),(10,-11,4),... since P_{5,5}=x^5, P_{6,5}=-5x^4+2x^6, P_{7,5}=10x^3-11x^5+4x^7,...

%p if modp(n-k, 2)=0 then a[n, k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-5, i)*binomial(n+k-5-2*i, n-5), i=0..(n+k)/2-5); end if;

%t DeleteCases[#, 0] &@ Flatten@ Table[(-1)^((n - k)/2) * Sum[(-1)^i * Binomial[(n + k)/2 - 5, i] Binomial[n + k - 5 - 2 i, n - 5], {i, 0, (n + k)/2 - 5}], {n, 5, 14}, {k, 0 + Boole[OddQ@ n], n, 2}] (* _Michael De Vlieger_, Jul 05 2019 *)

%Y Cf. A008310, A053117.

%K sign,tabf

%O 5,2

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