A308035 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - x + sqrt(1 - 2*x + (1+4*k)*x^2)).

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -3, 1, 1, 1, -3, -8, -3, 1, 1, 1, 1, -4, -11, 1, 21, 11, 1, 1, 1, -5, -14, 9, 61, 51, 15, 1, 1, 1, -6, -17, 21, 121, 91, -41, -13, 1, 1, 1, -7, -20, 37, 201, 101, -377, -391, -77, 1

Offset

0,14

Links

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

Formula

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

A(n,k) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,j) * binomial(n-j,j)/(j+1) = Sum_{j=0..floor(n/2)} (-k)^j * binomial(n,2*j) * A000108(j).

(n+2) * 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,  0,  -1,   -2,    -3,    -4,    -5, ...
   1, -2,  -5,   -8,   -11,   -14,   -17, ...
   1, -3,  -3,    1,     9,    21,    37, ...
   1,  1,  21,   61,   121,   201,   301, ...
   1, 11,  51,   91,   101,    51,   -89, ...
   1, 15, -41, -377, -1203, -2729, -5165, ...

Mathematica

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, 2*j] * CatalanNumber[j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

Crossrefs

Columns k=2..3 give (-1)^n * A091593, A308036.

Main diagonal gives A307947.

Cf. A000108, A306684, A307860.

Sequence in context: A362277 A367955 A273894 * A336479 A221131 A126886

Adjacent sequences: A308032 A308033 A308034 * A308036 A308037 A308038

Keyword

sign,tabl

Author

Seiichi Manyama, May 10 2019

Status

approved