A342129 - OEIS

%I #32 Apr 28 2021 02:05:38

%S 1,1,0,1,1,0,1,2,0,0,1,3,2,-1,0,1,4,6,0,-1,0,1,5,12,9,-4,0,0,1,6,20,

%T 32,9,-8,1,0,1,7,30,75,80,0,-8,1,0,1,8,42,144,275,192,-27,0,0,0,1,9,

%U 56,245,684,1000,448,-81,16,-1,0,1,10,72,384,1421,3240,3625,1024,-162,32,-1,0

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x + k*x^2).

%H Seiichi Manyama, <a href="/A342129/b342129.txt">Antidiagonals n = 0..139, flattened</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.

%F T(n,k) = (-1)^n * Sum_{j=0..floor(n/2)} (-k)^(n-j) * binomial(n-j,j) = (-1)^n * Sum_{j=0..n} (-k)^j * binomial(j,n-j).

%F T(n,k) = sqrt(k)^n * S(n, sqrt(k)) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

%e Square array begins:

%e   1,  1,  1, 1,   1,    1, ...

%e   0,  1,  2, 3,   4,    5, ...

%e   0,  0,  2, 6,  12,   20, ...

%e   0, -1,  0, 9,  32,   75, ...

%e   0, -1, -4, 9,  80,  275, ...

%e   0,  0, -8, 0, 192, 1000, ...

%p T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:

%p seq(seq(T(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 01 2021

%t T[n_, k_] := (-1)^n * Sum[If[k == j == 0, 1, (-k)^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 28 2021 *)

%o (PARI) T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));

%o (PARI) T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));

%o (PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));

%Y Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.

%Y Rows 0..1 give A000012, A001477.

%Y Main diagonal gives (-1) * A109519(n+1).

%Y Cf. A342120, A342133, A342134.

%K sign,tabl

%O 0,8

%A _Seiichi Manyama_, Feb 28 2021