A307968 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 + k*x + sqrt(1 + 2*k*x + k*(k+4)*x^2)).

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, 2, 0, 1, -4, 6, 4, -3, 0, 1, -5, 12, 0, -24, -1, 0, 1, -6, 20, -16, -63, 48, 11, 0, 1, -7, 30, -50, -96, 297, 24, -15, 0, 1, -8, 42, -108, -75, 896, -621, -464, -13, 0, 1, -9, 56, -196, 72, 1875, -3904, -1053, 1376, 77, 0

Offset

0,8

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 - k*x - k*x^2)^(n+1).

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

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

Example

Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,  -1,   -2,    -3,    -4,     -5,     -6, ...
   0,   0,    2,     6,    12,     20,     30, ...
   0,   2,    4,     0,   -16,    -50,   -108, ...
   0,  -3,  -24,   -63,   -96,    -75,     72, ...
   0,  -1,   48,   297,   896,   1875,   3024, ...
   0,  11,   24,  -621, -3904, -13125, -32184, ...
   0, -15, -464, -1053,  6912,  53125, 200880, ...

Mathematica

T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-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=0..2 give A000007, A007440(n+1), A307969.

Main diagonal gives A307946.

Cf. A107267, A307819.

Sequence in context: A292133 A304482 A309021 * A338501 A242464 A273185

Adjacent sequences: A307965 A307966 A307967 * A307969 A307970 A307971

Keyword

sign,tabl

Author

Seiichi Manyama, May 08 2019

Status

approved