|
|
A342129
|
|
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).
|
|
4
|
|
|
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, 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, 56, 245, 684, 1000, 448, -81, 16, -1, 0, 1, 10, 72, 384, 1421, 3240, 3625, 1024, -162, 32, -1, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
LINKS
|
|
|
FORMULA
|
T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) - T(n-2,k)) for n > 1.
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).
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.
|
|
EXAMPLE
|
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 0, 2, 6, 12, 20, ...
0, -1, 0, 9, 32, 75, ...
0, -1, -4, 9, 80, 275, ...
0, 0, -8, 0, 192, 1000, ...
|
|
MAPLE
|
T:= (n, k)-> (<<0|1>, <-k|k>>^(n+1))[1, 2]:
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(PARI) T(n, k) = (-1)^n*sum(j=0, n\2, (-k)^(n-j)*binomial(n-j, j));
(PARI) T(n, k) = (-1)^n*sum(j=0, n, (-k)^j*binomial(j, n-j));
(PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)/2));
|
|
CROSSREFS
|
Columns 0..10 give A000007, A010892, A099087, A057083, A001787(n+1), A030191, A030192, A030240, A057084, A057085(n+1), A057086.
Main diagonal gives (-1) * A109519(n+1).
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|