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A342120
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, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 16, 5, 0, 1, 5, 20, 45, 44, 8, 0, 1, 6, 30, 96, 171, 120, 13, 0, 1, 7, 42, 175, 464, 648, 328, 21, 0, 1, 8, 56, 288, 1025, 2240, 2457, 896, 34, 0, 1, 9, 72, 441, 1980, 6000, 10816, 9315, 2448, 55, 0
OFFSET
0,8
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) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n-j,j) = Sum_{j=0..n} k^j * binomial(j,n-j).
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.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 2, 6, 12, 20, 30, ...
0, 3, 16, 45, 96, 175, ...
0, 5, 44, 171, 464, 1025, ...
0, 8, 120, 648, 2240, 6000, ...
MAPLE
T:= (n, k)-> (<<0|1>, <k|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, 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 *)
PROG
(PARI) T(n, k) = sum(j=0, n\2, k^(n-j)*binomial(n-j, j));
(PARI) T(n, k) = sum(j=0, n, k^j*binomial(j, n-j));
(PARI) T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I/2));
CROSSREFS
Columns 0..10 give A000007, A000045(n+1), A002605(n+1), A030195(n+1), A057087, A057088, A057089, A057090, A057091, A057092, A057093.
Rows 0..2 give A000012, A001477, A002378.
Main diagonal gives A109516(n+1).
Sequence in context: A306704 A091063 A246935 * A198793 A085388 A351339
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Feb 28 2021
STATUS
approved