A307860 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).

1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -3, -5, 1, 1, 1, -5, -11, -5, 1, 1, 1, -7, -17, 1, 11, 1, 1, 1, -9, -23, 19, 81, 41, 1, 1, 1, -11, -29, 49, 211, 141, 29, 1, 1, 1, -13, -35, 91, 401, 181, -363, -125, 1, 1, 1, -15, -41, 145, 651, 41, -2015, -1791, -365, 1

Offset

0,13

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Formula

A(n,k) is the coefficient of x^n in the expansion of (1 + x - k*x^2)^n.

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

n * A(n,k) = (2*n-1) * A(n-1,k) - (1+4*k) * (n-1) * A(n-2,k).

Example

Square array begins:
   1,  1,    1,     1,     1,      1,      1, ...
   1,  1,    1,     1,     1,      1,      1, ...
   1, -1,   -3,    -5,    -7,     -9,    -11, ...
   1, -5,  -11,   -17,   -23,    -29,    -35, ...
   1, -5,    1,    19,    49,     91,    145, ...
   1, 11,   81,   211,   401,    651,    961, ...
   1, 41,  141,   181,    41,   -399,  -1259, ...
   1, 29, -363, -2015, -5767, -12459, -22931, ...

Mathematica

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

Crossrefs

Columns k=0..5 give A000012, A098331, A098332, A098333, A098334.

Main diagonal gives A307862.

Cf. A307819, A307855.

Sequence in context: A355815 A154512 A030588 * A123701 A143303 A074903

Adjacent sequences: A307857 A307858 A307859 * A307861 A307862 A307863

Keyword

sign,tabl

Author

Seiichi Manyama, May 02 2019

Status

approved