A342120 - OEIS

%I #37 Apr 28 2021 02:05:48

%S 1,1,0,1,1,0,1,2,2,0,1,3,6,3,0,1,4,12,16,5,0,1,5,20,45,44,8,0,1,6,30,

%T 96,171,120,13,0,1,7,42,175,464,648,328,21,0,1,8,56,288,1025,2240,

%U 2457,896,34,0,1,9,72,441,1980,6000,10816,9315,2448,55,0

%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="/A342120/b342120.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) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n-j,j) = Sum_{j=0..n} k^j * binomial(j,n-j).

%F T(n,k) = (-sqrt(k)*i)^n * S(n, sqrt(k)*i) 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, 2,   6,  12,   20,   30, ...

%e   0, 3,  16,  45,   96,  175, ...

%e   0, 5,  44, 171,  464, 1025, ...

%e   0, 8, 120, 648, 2240, 6000, ...

%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_] := Sum[If[k == j == 0, 1, k^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 28 2021 *)

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

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

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

%Y Columns 0..10 give A000007, A000045(n+1), A002605(n+1), A030195(n+1), A057087, A057088, A057089, A057090, A057091, A057092, A057093.

%Y Rows 0..2 give A000012, A001477, A002378.

%Y Main diagonal gives A109516(n+1).

%Y Cf. A342129, A342133, A342134.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Feb 28 2021