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A333989 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2). 2
1, 1, 1, 1, 0, 1, 1, -1, -4, 1, 1, -2, -7, 0, 1, 1, -3, -8, 23, 16, 1, 1, -4, -7, 64, 17, 0, 1, 1, -5, -4, 117, -128, -241, -64, 1, 1, -6, 1, 176, -527, -512, 329, 0, 1, 1, -7, 8, 235, -1264, 237, 4096, 1511, 256, 1, 1, -8, 17, 288, -2399, 3776, 11753, -8192, -5983, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).

T(0,k) = 1, T(1,k) = 1-k and T(n,k) = -2 * (k-1) * T(n-1,k) - (k+1)^2 * T(n-2,k) for n>1.

EXAMPLE

Square array begins:

  1,  1,    1,    1,    1,     1, ...

  1,  0,   -1,   -2,   -3,    -4, ...

  1, -4,   -7,   -8,   -7,    -4, ...

  1,  0,   23,   64,  117,   176, ...

  1, 16,   17, -128, -527, -1264, ...

  1,  0, -241, -512,  237,  3776, ...

MATHEMATICA

T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)

PROG

(PARI) {T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}

CROSSREFS

Main diagonal gives A333991.

Cf. A307884, A333988.

Sequence in context: A204456 A143441 A279206 * A016527 A010325 A265273

Adjacent sequences:  A333986 A333987 A333988 * A333990 A333991 A333992

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Sep 04 2020

STATUS

approved

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Last modified June 26 13:24 EDT 2022. Contains 354883 sequences. (Running on oeis4.)