A342134 - OEIS

%I #32 Apr 27 2021 02:11:43

%S 1,1,0,1,2,0,1,4,5,0,1,6,18,12,0,1,8,39,80,29,0,1,10,68,252,356,70,0,

%T 1,12,105,576,1629,1584,169,0,1,14,150,1100,4880,10530,7048,408,0,1,

%U 16,203,1872,11525,41344,68067,31360,985,0,1,18,264,2940,23364,120750,350272,439992,139536,2378,0

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

%H Seiichi Manyama, <a href="/A342134/b342134.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) = 2*k and T(n,k) = k*(2*T(n-1,k) + T(n-2,k)) for n > 1.

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

%F T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.

%e Square array begins:

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

%e   0,  2,    4,     6,     8,     10, ...

%e   0,  5,   18,    39,    68,    105, ...

%e   0, 12,   80,   252,   576,   1100, ...

%e   0, 29,  356,  1629,  4880,  11525, ...

%e   0, 70, 1584, 10530, 41344, 120750, ...

%p T:= (n, k)-> (<<0|1>, <k|2*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, (2*k)^j] * 2^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 27 2021 *)

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

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

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

%Y Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).

%Y Rows 0..2 give A000012, A005843, A007742.

%Y Main diagonal gives A109517(n+1).

%Y Cf. A342120, A342129, A342133.

%K nonn,tabl

%O 0,5

%A _Seiichi Manyama_, Mar 01 2021