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A342133
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).
4
1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0
OFFSET
0,5
FORMULA
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.
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).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
0, 3, 14, 33, 60, 95, ...
0, 4, 48, 180, 448, 900, ...
0, 5, 164, 981, 3344, 8525, ...
0, 6, 560, 5346, 24960, 80750, ...
MAPLE
T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021
MATHEMATICA
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 *)
PROG
(PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));
(PARI) T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));
(PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));
CROSSREFS
Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
Rows 0..2 give A000012, A005843, A033991.
Main diagonal gives (-1) * A109520(n+1).
Sequence in context: A138752 A357499 A368506 * A358050 A334781 A291656
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Mar 01 2021
STATUS
approved