A306913 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k+x^k).

1, 1, -2, 1, -2, 4, 1, -3, 2, -8, 1, -4, 6, 0, 16, 1, -5, 10, -11, -4, -32, 1, -6, 15, -20, 21, 8, 64, 1, -7, 21, -35, 34, -42, -8, -128, 1, -8, 28, -56, 70, -48, 85, 0, 256, 1, -9, 36, -84, 126, -127, 48, -171, 16, -512, 1, -10, 45, -120, 210, -252, 220, 0, 342, -32, 1024

Offset

0,3

Links

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

Formula

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

Example

Square array begins:
      1,  1,    1,    1,    1,    1,     1,     1, ...
     -2, -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
      4,  2,    6,   10,   15,   21,    28,    36, ...
     -8,  0,  -11,  -20,  -35,  -56,   -84,  -120, ...
     16, -4,   21,   34,   70,  126,   210,   330, ...
    -32,  8,  -42,  -48, -127, -252,  -462,  -792, ...
     64, -8,   85,   48,  220,  461,   924,  1716, ...
   -128,  0, -171,    0, -385, -780, -1717, -3432, ...
    256, 16,  342, -164,  715, 1209,  3017,  6434, ...

Mathematica

A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

Crossrefs

Columns 1-2 give A122803, A108520.

Cf. A039912, A306914, A306915.

Sequence in context: A110162 A350228 A199087 * A087704 A165092 A306915

Adjacent sequences: A306910 A306911 A306912 * A306914 A306915 A306916

Keyword

sign,tabl

Author

Seiichi Manyama, Mar 16 2019

Status

approved